Alright - so we just addressed the issue, by restarting SQL Server.

Not exactly the solution that I'd like to rely upon though, especially since this is a production server.

One thing we noticed is that the server is running on a 64-bit machine, but the machine only has 4GB of RAM, of which 3ish GB was being used by SQL Server. Upon restarting, it went down to around 150MB.

Now, we haven't noticed any performance problems from the fact that it has only 4GB of RAM, and I can't imagine why a shortage of RAM might impact the communication between the server and one website, while leaving the communication between the server and another website unaffected.

If anyone can help me find the problem that was there, and address it, I'd greatly appreciate it, as we plan on having other CLR processes running in the future, the importance of which will be far greater than that which we have currently running.