Afghan officials had said earlier two British soldiers were wounded in the incident.
The U.S. military said in a statement the attacker died in the morning attack in Lashkargar, the largest city in Helmand Province.
The bomb went off outside the main gate of the base of a British-run Provincial Reconstruction Team. No one has yet claimed responsibility for the assault.
Some 3,300 British troops are currently being deployed in Helmand, one of Afghanistan's main opium-producing regions.