Thanks for the kind words and suggestions, everyone. We went through with it yesterday (saturday) afternoon.
I'm very glad we decided to do it at our home. He was able to be comfortable, surrounded by the people who loved him and in a familiar place. It went without a hitch, and all things considered, I think that it went as well as it could.
The other dog was kept in another room and didn't see him after. But she didn't eat all day today, so she obviously knows that something's up. Thanks again everyone.