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.