I've done it (see my previous post).
I reckon if you're just going for a month just go on the visa waiver scheme? That lets you stay there for 3 months I believe, no visa required; of course it's supposed to be for tourism only.
Otherwise, you will need a J1 Scholar visa. There's 2 types:
1) J1 research scholar - This is good for 1 year, with the possibility for extension for up to 5 years total. Once you return to the UK, you would not be able to come to the US on a J1 visa for 2 years (I think this is negotiable though)
2) J1 short-term scholar visa - This is good for 6 months maximum and cannot be extended. However, there is no time restriction on returning to the US on a J1 visa afterwards.
Check this website, as my info may be out of date: https://j1visa.state.gov/. Plus I was getting paid by the US uni and it might be different if you're not.
For me, the university I was going to paid for the visa, otherwise it's about £600 I think - there's stuff for them to pay and stuff for you to pay. You have to go to the US embassy to get the visa. The process started in May and I went to the US in July. Visa turn around is a few weeks I think. Pretty forgone conclusion you would get it.
This website also has useful info: http://www.fulbright.org.uk/
I reckon go for longer than a month - you'll need it.