Sharon John and Patrick Mullane, the parents of six-day-old Eliza-Mae who is thought to have been killed by the family dog, today said she "will always be in our hearts and thoughts and we will cherish the little time we were able to share with her".
Eliza-Mae Mullane was airlifted to the University of Wales Hospital in Cardiff yesterday after what is thought to have been an attack by the family's Alaskan malamute dog.
In a statement, 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.
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"She was a dearly loved daughter, sister, granddaughter and niece.
"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.
"We are extremely grateful for all the support shown to us by the community. We are now trying as a family to come to terms with what's happened, and ask that we are allowed to do this in privacy. Thank you."
It is "very unlikely" that there will be any updates on the police investigation until the end of the week, a spokeswoman for Dyfed Powys Police said.
A post-mortem examination is not expected to be carried out until then.
The dog is still alive, and the spokeswoman said: "We've still got the dog, and no, it hasn't been destroyed at the moment."
The parents have not been interviewed by police, she added.
The force has not confirmed whether the animal was responsible for the death, or the type of injuries sustained by Eliza-Mae.
Neighbour Patricia Punter, 70, has been widely reported as overhearing the baby's mother screaming: "The dog ate my baby's head."
Police were called to New Road, Pontyberem, Carmarthenshire, Mid Wales, at 8.30am yesterday, when officers seized the dog.
Alaskan malamutes are not listed under the Dangerous Dogs Act and police have said the investigations are still at an early stage.
A police spokeswoman said yesterday: "We are not going to speculate on reports from people in the community and we respectfully ask that you wait for the investigation to run its proper course."
The Alaskan Malamute Club of the United Kingdom describes the breed as an "affectionate, friendly, loyal, devoted companion" on its website.
An ideal Alaskan malamute is "heavily boned, powerfully built", it says.
The death comes barely a week after 11-month-old Ava-Jayne Corless was killed by a dog as she slept.
The girl was in bed at a house in Blackburn last Monday when she was savaged by a pit bull terrier.
The girl's mother Chloe King, 20, and her partner, Lee Wright, 26, were held on suspicion of manslaughter and subsequently bailed.