Poppy was born in New York yesterday and has been named after Jenna's grandfather, former US president George H W Bush, who was nicknamed Poppy as a child, he said.
"Laura and I are overjoyed to meet our second grandchild today," said Bush, who was president from 2001-2009.
"We are grateful to God -- and to Jenna and Henry -- for this beautiful, healthy gift of life."
Jenna Bush Hager, 33, welcomed her first daughter, Margaret Laura "Mila" Hager in April 2013. She is the younger of Bush's twin daughters and works as a correspondent on NBC's "Today" show.