"I've had a feeling this whole time it was going to be a girl," said Joshua on the Today show before learning the baby's sex.
"I think it's a boy, but a girl would be great," added Anna.
The couple then learned the sex of their baby after Joshua cut into a cake that revealed pink layers.
Joshua and Anna were married last September -- only a few months after Jim Bob and Michelle welcomed their eighteenth child, Jordyn-Grace Makiya Duggar. Joshua and Anna are expecting their new addition to the Duggar family in October.
"I'm feeling great," said Anna on the Today show. "We've been able to feel the baby move and kick, so that's been a lot of fun."
While Joshua is already an older brother for 17 siblings, he acknowledged that being a father is "different than just being an older brother" because it comes with "a lot more responsibility."
"I think you can prepare only in so many ways, but really just realizing that you're going to be a role model for that child, and be able to be prepared as you can be to be able to take on that," he said on Today. "It's an interesting experience, but I have great parents to look to."
Jim Bob said he and Michelle are "really excited" about becoming grandparents when he appeared via satellite during the Today show appearance.
"We've been dreaming about having grandbabies and now it doesn't even seem possible that we're at this point," added Michelle. "It's so exciting, and we are thrilled for Josh and Anna. They're going to be great parents."