Thanks for the replies.
Unfortunately, changing one of the subnets didn't work. However, after some experimentation, I've found a solution. Basically, you have to "bridge" the two networks together. I'm not sure whether bridging is the best, or even the only solution. But is seems to work. Now everything is working beautifully.
ipconfig now shows the following:
Ethernet adapter Network Bridge (Network Bridge):
Connection-specific DNS Suffix . :
IP Address. . . . . . . . . . . . : 192.168.1.6
Subnet Mask . . . . . . . . . . . : 255.255.255.0
Default Gateway . . . . . . . . . : 192.168.1.1
The above settings were generated automatically by my router.
The only possible problem I can see is that the router's DHCP system appears to reserve IP addresses from 192.168.1.2 to 192.168.1.33. So to be on the safe side, I might change the Xbox's address to 192.168.1.34.