Can I play UK (region 2) DVDs on a US (region 1) Xbox 360?

No, a region 1 Xbox is like any other DVD player - it will not play DVDs from other regions:

Each Xbox 360 console is manufactured for a specific DVD and game region, following international standards. The Xbox 360 console can only play discs that are sold in the same region as the console.

Your only option (short of modding the console, which you probably don't want to do if you play online) is to get a region-free DVD player or a region-free DVD-ROM and rip your DVDs into a video format that the Xbox recognizes to play them from an external drive or streamed from a computer.

Some computer DVD-ROM drives allow you a certain number of region changes, so you may be able to do that with the drive already in your computer (assuming you have one :)) instead of spending money just to convert this one DVD.

I realize this is an old post, but for future askers of this question, on most DVD players regions can be changed. There is no need to buy a new DVD player. You can program some to accept any region with a certain programming code, sometimes using region 0. Do the Internet search for the DVD player model number and keywords “region change”, and the instructions will usually come up. Usually a sequence of button presses on the DVD remote.

This makes sense — manufacturers aren’t going to make each DVD player specific to a region; they simply program them as such. I’ve changed several DVD players to do this, because I used to buy DVDs on eBay, often with a different region code.