What word or expression can I use for a baby who is the exact copy of one of its parents?
One such phrase is the spitting image.
Like others have said, the spitting image and dead ringer are good ones to describe family members who look exactly alike. Carbon copy is my personal favorite. "He is a carbon copy of his dad!"
You could perhaps use duplicate, look-alike, twin or doppelganger depending on what you're going for. Some words would work better for strangers rather than relatives though (ex. "you two could be twins!" or "she's his doppelganger, I swear" or "she's an exact duplicate of him").
Here are some synonyms for spitting image that you might find useful.
Maybe this is quite recent, but many people will recognize Mini-Me as an exact (although smaller) copy of someone.