"We are so grateful to God for the arrival of little Samuel Scott," Jill Dillard wrote on her Instagram account.
The couple have shared more photos of their new baby on their blog, "The Dillards."
Samuel has one older brother, Israel, who will turn 2 in August.
Jill is the fourth of the Duggars' 19 children, and appeared on their original TLC TV show "19 Kids and Counting." The series was suspended, then canceled, by TLC after the Duggars' oldest son, Josh, publicly apologized for molesting five girls, including several of his own sisters, when he was 14 and 15 years old.
Jill now appears on "Counting On," the TLC show created in 2015 after "19 Kids and Counting" was canceled.
Jill's sister, Jessa Duggar Seewald, gave birth to her second son, Henry, in February. Their brother Josh and his wife Anna are expecting their fifth child later this year.