The victim, identified as Sally Jane Adey, died along with three Tunisians and tourists from Japan, Italy, Colombia, Australia, Spain, France and Poland, in the attack at the Bardo Museum in Tunis.
"We will do everything we can to help the family of the British woman murdered in Tunisia. We will not let terrorists undermine democracy," Cameron said.
UK foreign secretary Philip Hammond, who is in Nairobi, told the BBC that British consular staff were helping Adey's family.
"I spoke to Tunisian Prime Minister Habib Essid yesterday to offer my condolences to all those affected by this despicable act and to assure him we stand together against terrorism," he said.
Officials in Tunisia said that more than 40 people, including tourists and Tunisians, were injured in the attack.
Security forces have killed two gunmen but a hunt was still on for their accomplices.
