When I went to check on the baby goats this afternoon, I found that we had lost one. She was lying, limp, in the shelter. We immediately brought the other baby into the mudroom. So far, we've gotten some milk replacer in her and she's moving around and letting us know, rather loudly, that she's not too sure about the new set up.
I'll keep you posted on our progress.
Update: The surviving baby seemed fine after we got some milk in her. She's producing urine (a sure sign that milk is getting through) and one of our "goat friends" said she should be okay to go back to her mother for the night. We'll keep supplementing with milk replacer until she's a bit stronger.