making him STAY!

Have you tried giving him treats while he's in a down (so he doesn't have to get up)? e.g. getting down to his level, or throwing them down by his nose.
 
Do you have a no-reward marker? What I would do is if he breaks the down or sit before you release him is just give him a NRM (and no reward...[8|]), and let him try again.
 
With Pocket, I gradually increased the length she had to stay down before I gave her a treat (for instance, 10 correct 1s downs, then increased to 2s, 3s etc...).
After she was able to hold a down for a relatively decent amount of time, then I began taking one small step before giving her a treat (if she didn't move), then gradually increasing the size/number of steps required for a reward. If she broke the down/sit before I released her, I put her back where she was, and started again.
 
I think the most important thing with teaching a dog to stay/hold a position is to take baby steps and not too expect too much too soon. Also sometimes it's easier to do stays with a down first, because it's a lot easier to break a sit.
 
Good luck! You'll get there!

Ditto everything ^^^^
 
Can't emphasise enough baby steps baby steps baby steps. 
 
The down is considerably easier to teach stay in, IME. 
 
I also try to make sure he gets the reward while he is in that position rather than after he has got up.... you can sit beside him and use a piece of furniture or the crook of your leg so he has to stay down to take the reward.  I reserve some small yummy chewy rewards for teaching down/stay too, because i put them between the front paws and they stay in the right position while munching, creating a good habit and a good association.
 
I don't say "Stay" initially, because there's the risk the dog hasn't "got it" yet and will break from the down before I release him, I wait until the stay is quite solid before I introduce the word, then I make sure I go back to the dog to reward him for staying before he looks like he will get up.  I rarely call him from a "stay". 
 
When you get to the point where you start moving away, I've found realistically you normally want the dog to "stay" as you walk away from them so I avoid walking away backwards while teaching it.  (I know loads of people who teach stay like this and the dog just gets up to follow you if you turn your back....)
 
What I do is I put the dog "at heel" and then get a down, then I step away with the foot furthest from his face and just transfer my weight to that foot before bringing it back and rewarding the dog.  Next time I step away properly and back, next time I take a couple of steps, pivot and go back and so on. 

Start small and build up gradually! Here is the stay exercise my trainer used:

1. Have dog sit or lie down.
2. About three feet away, place treat.

Your dog will probably get up to go and get the treat as you are putting it down. Just pick the treat back up and ask dog to sit. She calls it the Hydraulic Butt Exercise. Butt down = treat moves down toward ground. Butt up = treat moves back up off the ground.

3. The split second your dog has enough self-control to let the treat hit the floor while his butt is on the ground, tell him he can go and get it (I use "Get it!")

4. Repeat, repeat, repeat, increasing the duration between the treat hitting the floor and your "get it".

Tips for success:

1. don't expect too much. Increasing the length of the "stay" by a second or two at a time, watching your dog for fidgeting, is better than going right to 5 minutes.

2. Practice like a million times a day. Once he's got it down, have him "wait" to go out the door, to get his dinner, drink water, get petted, etc.

3. Choose one variable at a time to work on. Example: if you increase or decrease the distance between the treat and the dog, then don't increase the duration too.

Good luck!

It has occurred because the word has been used and the concept not learned.  He literally believes that he is doing the right thing.  There is a whole constellation of cues in your tone, body position and movement.  There is a point at which, he sees from you a signal that he has learned to mean  "stay is over".   Some options:
 
video tape you and examine what you do, watch closely the interaction between your actitions.  That will help you identify the elements to change in your behavior.
 
another option.  Get your dog in position, hold in your hand a treat.  Hold the treat at nose level, close to the dog.  If the dog breaks position, close your hand and WAIT until the dog returns to the position (you may have to position occassionally in the first trials).  When the stay has been held (5 seconds maybe), say good stay and allow the dog to eat the treat WHILE staying in the position.
 
another option, place the dog in position,  give stay command.  With the leash attached to the collar,  place your hand close to the dog's body to prevent movement.  Say "stay"  apply slow gentle pressure with the leash straight out from the collar,  vary directions once the dog has become accustomed to the sensation of the leash pressure.  You MUST have your hand ready to prevent position change when the dog feels the leash pressure.
 
From Leslie Nelson,,, make the release from the stay VERY low key,  All the attention and reward should occur while the dog is in the stay position maintaining the "stillness". 

Ditto everything above and...

I informally reinforce the value of staying put by requiring the dogs sit or lay down and stay until released for a wide variety of things they want and can tangibly see coming nearer or further to them as they either stay or get up (a la the "hydraulic butt" exercise above). For their food dishes, they must lay down until released. Dogs tend to catch on to this very quickly IF you make it easy for them to succeed the first few times (meaning, don't try to make them stay for a full minute the first time, just a few seconds) and you don't give up and give in if they aren't getting it. They won't get it until you absolutely pick that food bowl up every single time they try to get up from their position. The firmer you can be about that, the faster this will go. You have to have some quick relfexes and be prepared here. Set the food bowl down very slowly and if the dog begins to move, pick it up again. Once you're able to get it on the ground, again if they move pick it up again. But once you've got it on the ground and they haven't moved, give your release word quickly so they can succeed in the task with a minimum of frustration. As they do this more and more you can extend the time they're staying.

I do the same exercise for going in and out of doors to the outside or in and out of the baby gate. That's about 6 times or more a day of informal training opportunity for "stay until released' using things the dog can see clearly as related to their ability to stay.

Also remember the "three D's" of stay: distance, duration and distractions. The best bet is to only work on one "d" at a time, so first work on the distance you can move away from the dog as they maintain their stay, then work on the duration of the stay without adding in distance again until the duration is solid, then slowly add distractions but again don't worry about distance and duration until working with distractions is solid.

You've gotten excellent advice - the stay is a behavior that should be built slowly.  Do not expect much distance or length of time at first. 
Check out the video at this site to see what I mean:
www.clickertrainusa.com

You didn't mention if you taught your dog to "stay". Staying is tough for some dogs but you must be persistent. When your dog is sitting, tell him to "stay", put your open hand in front of his face. If he stays for even a second, praise him (not too exictedly) and give him a little tasty treat. You don't want him to fail at this or any other command so when he makes any kind of progress at all you must acknowledge it. You will want to distance yourself from him a little at a time but do this slowly and don't over work this command. Training him to do anything will be easier if you exercise him first. Exercise, train, play. In that order.

I trained stay in conjunction with my release command ("OK").  At first I have the OK immediately after the stay command, and then very gradually lengthened the time between the command and the release.  So at first I had "sit" then immediately "stay" (with a hand signal) then immediately "OK".  Then I started waiting a little between the stay and the OK, adding in a "good stay" as he stays still. You want to offer treats while he is still staying -- otherwise he might think that he's getting the treat for getting up.  If he starts to get up before he hears OK, use a "eh eh" type of monor correction and then begin again with stay and OK.  You need to gradually lengthen it out -- if he's getting up you are probably moving too fast.  If that happens, give him another, easier chance to get it right and move from there. 
 
Once he has the idea, practice like crazy and make it more challenging as he gets better -- get up and walk away from him, leave the room, get out some treats -- the key is that your "OK" and nothing else ends the stay.  Once he has an out-of-sight stay, playing "find it" (where you hide a toy or treat while he stays, then you release him to go find it)is a great way to pratcice and reinforce his skills. 

Thanks for all the suggestions! mrv- I think the idea of videotaping is excellent. And thanks for the link, spiritdogs! I have been reading a limited amount concerning clicker training, but I have yet to "get into" it.

Training him to do anything will be easier if you exercise him first. Exercise, train, play. In that order.

I guess it depends on the dog, but FWIW, I've found that after exercise my dog is tired and less likely to be able to focus.