Thanks for all the replies.
Snaake: from what I've been reading that appears to be the converter to get. However, it's a bit expensive. In fact it costs almost as much as I paid for my (new) Xbox plus two games! I'm trying to keep the cost of this project down so I think I'll explore other alternatives first.
Also, it won't work with PAL Xboxes and I'm in a PAL region (I'm not sure whether modding will get round this problem).
I spent hours reading up on this yesterday, and so far I've found out the following:
The Xbox does indeed appear to be able to generate displays at a resolution and refresh rate that a VGA monitor can handle so scan conversion shouldn't be necessary. However, you have to recreate the VGA sync signals. There are several different approaches to doing this. Each has their pros and cons.
I've come across several reasonably simple circuits on the net for restoring the sync signals. This really shouldn't be necessary but it appear that Micro$oft, in their infinite wisdom, deliberately removed the VGA functionality from what is essentially a standard PC graphics chip.

You need to mod your Xbox but there is a question mark over whether 1.6 Xboxes are yet supported. I'm not sure whether you need to mod the Xbox to get in-game VGA support or whether it's just to enable the startup screen to be in VGA. The sites I've looked at are unclear on this point.
The best links I've found so far are as follows:
http://antibeat.host.sk/vgaguide/http://xbox-scene.com/articles/vga-cable.phphttp://www.360mods.net/News/article/sid=32.html