iPhone in endless recovery loop

My 4S iPhone is in an endless loop trying to install iOS 5.1.1. I plugged it into iTunes like usual. I told it to download the new software. It did that but now iTunes tells me the phone is in recovery mode and I must restore. I do that and it goes through all the steps but ends up with the same recovery mode message. Over and over again.

I used time machine to get an earlier backup. Same story. Putting the phone into DFU mode (whatever that stands for) did not help.

Now what?

I have Lion 10.7.3 and iTunes 10.6.1.