There's some misdirection that implies the fate of the baby, which you'll get to later on as you progress through the show. This makes it a little hard to give a straight yes/no answer to your question, since you request no spoilers. If you do want it spoiled:
This interview with the show's creator explains the baby's fate. It's implied that Peggy's sister maybe takes the baby to raise it. But that's not the case, the baby is put up for adoption, and we never see it again.
There is some dialogue later on that does discuss the baby and what happened to it. So, this plot thread didn't just get dropped, it does return.