A homicide investigation has been launched following Hope's death at Auckland's Starship Hospital on Wednesday.
Schultz said Hope was taken to Southland Hospital in Invercargill just after midnight on Friday, after she stopped breathing at her Appleby home.
Her mother, Brittany, had been out visiting and received a phone call urging her to come home immediately, Schultz said.
The infant was able to be resuscitated at Southland Hospital, but did not regain consciousness and was transferred to Starship in a critical condition.
The decision was made to turn off her life support on Wednesday evening, and Hope died in her mother's arms.
Schultz said she was very close with her stepdaughter and all the family was still struggling to come to terms with what had happened. Hope had been a "surprise" baby in several ways and her personality was just starting to blossom after a tough start.
The family had first been a bit shocked when tomboy Brittany said she was pregnant, then "really excited", Schultz said.
"When she told me she was having a girl, I was super-excited."
Apart from the usual struggles premature babies had, Hope had been doing really well.
One visit was still fresh in Schultz's mind.
"[Hope] was making noises and giggling. I said to Brittany, 'look, she has a big smile on her face'."
"I just absolutely love my grandkids. I love having them around. Now we are not going to have that opportunity to have Hope around."
Hope's grandfather, Chris, was also "absolutely devastated".
"Now I know how all these other mothers and grandmothers feel when we hear [things] on the news. Now I actually know how they feel. It's horrible."
The double tragedy was that Brittany, 22, had been told she would not be able to have any more babies.
"The baby was absolutely her world. Her everything.
"She loves working with kids. Especially intellectually handicapped people. She loves helping people."
Police yesterday said officers had spoken with others at the Elizabeth St property and were continuing to speak with the street's residents.
An ESR team assisted police with a scene examination at the house earlier in the week.