This might work for you. A few years back I had gotten laid off due to budget cuts. Despite the fact that had been laid off 2 weeks into the job and that I actually had proof and a letter from my employer that it was a budget issue I still had a SUPER hard time getting a job. I knew I would have to get some employment that showed I was stable.

So one day when I was at a small employer for an interview... (small because sometimes large companies will not do this) I came up with an on the spot idea. I said, look I have a proposal for you. I will work for you for 6 weeks for free. If you don't like my work, all I will do is ask for a reference and you can say the 6 weeks was a temp job. If you do like my work, you can hire me.

He went for it and he hired me.

In the right circumstance this can work. I suspect that he hired me mostly because it would have been such a hassle to go back to the search for someone. But if they don't want to keep you -- you can STILL list it as a job and they probably will provide you a good reference.