Miss Beazley (search), as the first lady and twins Barbara and Jenna have already named the pup, was born Oct. 28. She is due to arrive just before Christmas.
The dog was named for the character Uncle Beazley, a dinosaur in Oliver Butterworth's children's book, "The Enormous Egg."
Barney's half brother is Miss Beazley's father, said White House spokesman Scott McClellan.
Mrs. Bush turned 58 on Thursday.
The Bushes lost a longtime pet earlier this year. Spot, an English springer spaniel who regularly ambled around the West Wing with Barney, died in February at age 15.