Six-day-old Eliza-Mae Mullane was airlifted to the University of Wales Hospital in Cardiff on Tuesday after what is thought to have been an attack by the family's Alaskan malamute dog.
Her parents, Sharon John and Patrick Mullane, said: "Even though she was an important part of our family for such a short period of time, Eliza-Mae will always be in our hearts and thoughts and we will cherish the little time we were able to share with her.
"She brought joy into our family, and losing her like this has cast the most horrible shadow over all of us. There are no words we can use to describe what we feel at the moment, and we don't think there ever will be."
A spokeswoman for Dyfed Powys Police said: "We've still got the dog, and no, it hasn't been destroyed at the moment." The force has not confirmed if the animal was responsible for the death.