I would say that you take it as a life lesson, and be more scrupulous about preserving your integrity in the future. I would also say it's safe to assume that you've blown your chances of getting this job.
Honest people aren't people who are never dishonest, they are people who regret their moments of dishonesty, and are not dishonest in quite the same way again. By admitting that you haven't been entirely truthful, you've made a step in the direction of being an honest man, and a step away from the direction of being a dishonest one.
Now, you'll need to ask yourself why, and you probably ought to spend some time soul-searching before you try to explain it to the employer. (Things like being scared of not being able to find a job come to mind.) When you do attempt to explain yourself to the employer, you might start by saying that you recognize that you have blown your chances of getting hired, but that you wanted to speak on a personal level. Then pretty much bare your soul, in 100 words or fewer. (Not to try to get the job back, but to get it out of your system.) Then apologize and leave it at that. (You could thank them for the life lesson, too, if you can be sincere about it.)
(EDIT: Eric Lippert has pointed out that there are laws in some states that criminalize lying about your education on a resume. I would suggest that you get a lawyer's advice before telling the employer that you did this.)
Finally, you can be glad that you got caught now, instead of two or three years in. If you are an honest man, it would have eaten at you, and you would feel relieved when it finally came out in spite of the fallout. If you aren't an honest man, you'd probably find that you were unable to talk your way out of it, assuming you hadn't been fired already for some other lapse of integrity.
So, man up, face the consequences, forgive yourself (even if other people don't), do your best to put it right, and don't do it again.