Well, as far as I know, unless your diagnosis is well known to others, then you'd probably be able to lie and get in. I mean it's not like there's a database of people with Aspergers diagnosis that the US government is keeping somewhere that the military looks at. I think the only thing would be, if you knowingly went in with an Aspergers diagnosis, you might get in trouble if things went south during your time in the Army, and if you were discharged, it might be dishonorable discharge or loss of benefits or something to that effect.
I mean the Army already will, say, exclude people with depression or anxiety diagnosis, which many more people are diagnosed with than Aspergers, I'd guess lots of people in the Army probably got a middle school diagnosis of depression, too. I think even if the Army got access to your medical records (and in the case of, say, a therapist you paid cash for, this might be kinda hard) you'd then just have to see a military psychologist and be rediagnosed. So, excluding Aspergers isn't really like, that odd if you think of it that way.
Interesting what you said about the draft, auntblabby. Never knew it was quite like that. I do know Jimmy Hendrix got out of the draft by pretending he was gay in Basic, and Ted Nugent (who was totally pro-Iraq war, gee) got out of the Vietnam draft by sh*****g and pissing in his pants for 2 weeks before going to the recruiting office to make himself look crazy. As far as Aspies in a draft, though, it'd depend on a lot of things. I kinda wonder too, if some Aspies could make, say, a good sniper or really specialist combat role like that, not just a boring computer tech or whatever. In a draft, we'd at least have to cook food, haha.