Stephen Baber was buried in the north cloister of Westminster Abbey on 27th March 1679 but his gravestone is no longer readable. A record was made of the inscription:
Here lieth the body of Stephen Baber Gent. only son and child of Benjamin Baber, Gent, one of the Aldermen of the City of Bath by Elizabeth his wife, and was born October 19 1663 and departed this life March 23 1679
His father Benjamin was a son of Richard Baber of Tiverton in Somerset and was three times mayor of Bath (buried in Bath Abbey). His mother Elizabeth was daughter of William Fox of Farley in Wiltshire (probably a brother of Sir Stephen Fox).