I should know better, especially because the Apple Genius told me to do this, but I was so preoccupied with other features...
If you have upgraded from an older iPhone, and done a Restore using iTunes, FaceTime will NOT be activated even though it is active by default on a new iPhone. The act of restoring turns it off.
To remedy this, go to Settings -> Phone and toggle the switch for FaceTime.
Now I'm ready for my closeup!