Hmm, tough position to be in. One of the things you need to do is to get to know your professors for situations just like this. Supposedly that is the advantage of a small university, but in reality some people at the big universities get to know their professors as well.
Have you done any work with them or any TA's / lab instructors / research assistants? I got one or two rec letters from professors who didn't know me, but their TA's did. I told the professors what I was doing and that I've worked on their research with the TA's and I wanted to see if they would be willing to provide me a recommendation.
I would schedule an appointment to sit down with your advisor and explain the situation and how to approach it. I wouldn't directly ask him for a letter, but maybe they will offer (though I doubt it). You have only seem him once, but the semester is very early, so there is nothing wrong with that.
When is your application due anyway? If you have a month or two, I would take some time to actually get to know some of your profs. Explain that the prospect of the internship made you wake up and realize that you don't know anybody and would like to get to know some of them and get involved with what they do so that you can get a better learning experience and build a network. There's nothing wrong with that.
The other piece of advice I have is once you set these relationships up now, don't abandon them unless you are willing to abandon that reference permanently. It annoys me when I have somebody that was in one of my classes ask for a a rec letter and quickly become my best friend and biggest fan, and then never hear from them again (including whether they got into the program they were applying for). Or, just as bad, somebody who 2-3 years down the road, after no contact whatsoever, asks me for a reference. Either stay in touch professionally, or form new references... Don't expect to get a glowing reference from somebody who barely remembers who you are!
Good luck!