Apple caught the problem and e-mailed me to ask if it was me. I told them no. They disabled my account, refunded the gift card money, and asked me to write them once I was satisfied that my computer was secure (just in case it was key-logged [I was not].)
I wrote them back the next day and told them everything was good to go. They enabled my account and I signed in and changed my password.
I didn't lose any of my purchased items and I could have had the account back the same day had I chosen to.
I recall a while back that there were quite a few iTunes accounts that had been compromised. I used a very non-trivial password, too, so I'm rather surprised that mine was one of the compromised accounts. I'm still curious as to how it happened.