the error is quite clear (: you ve changed sth in your index.php and left a < outside or inserted a blank space somewhere by mistake.. we need to see that line and around it to actually tell you what u need to do

Neither me nor the other admin have touched anything, though. Could it have happened some other way?

via FTP, check your index.php file by comparing it to an original index.php from the downloaded package.
If it's been altered/hacked/messed up, you should be able to note the difference and edit it, or upload a fresh new one.

if you havent made any changes just replace it with the one in your latest backup.. if you dont have one, it'd be a fine warning for you to take backups before worse happens

Thanks for the help. We replaced the index file, and aside from a few corrupted skin files, everything seems to be working well.