Born in Kent County, Delaware, eight miles northeast of Greensboro, Maryland, Cubreth attended the public schools and studied under private tutors.
Culbreth was elected from the sixth district of Maryland as a Democratic-Republican to the Fifteenth Congress and reelected to the Sixteenth Congress, serving in the United States House of Representatives from March 4, 1817, to March 3, 1821.
He was appointed chief judge of the Caroline County orphans’ court in 1822 and was clerk of the executive council of Maryland from 1825 to 1838.
He resided in Annapolis, Maryland before returning to Denton in 1838 to work in mercantile pursuits.
Soon afterward, he moved to Orrell Farm, near Greensboro, where he died on April 16, 1843.