Neighboring countries have promised to send troops if Jammeh refuses to hand over power.
Barrow made the plea in a Christmas message, in which he pointed out that even Great Britain as a colonial power handed over the reins of the country peacefully.
Jammeh, who has been in power for 22 years, originally accepted the results of the December 1 election, but quickly reversed his position and now says he won't go unless ordered by the country's Supreme Court. But since Jammeh fired several judges earlier in the year, the body does not have a quorum to sit.
In his broadcast statement, Adama said, "I should assume office when the term of office of the incumbent expires… I do not want to preside over a country that is not at peace with itself." Jammeh's term is scheduled to end January 19.
The Economic Community of West African States (ECOWAS), of which The Gambia is a member, has condemned Jammeh's refusal to budge. The organization sent a delegation to try to talk sense to him, but were rebuffed. Now,
ECOWAS has named Senegal as the leader of any military operations needed to oust Jammeh, the BBC reports.
Adama alluded to this in his message. "Common sense and the supreme national interests dictate that we put our differences aside and accept the verdict of the people," he said. "I would urge all Gambians to work together to take full charge of our destiny and resolve our differences in peace, failing which, outsiders will determine our destiny for us, to our shame and disgrace."
The Gambia is surrounded on three sides by Senegal.
ECOWAS Chairman Marcel Alain de Souza said on December 23 that though ECOWAS does not want a conflict, "If [Jammeh] is not going, we have stand-by forces already alerted and these stand-by forces have to be able to intervene to restore the people's wish," he said.
"If he loves his people, he has to be able to negotiate an exit door calmly. If it doesn't happen, the most radical means will be used."
There are reports that "rogue soldiers" loyal to the sitting president have been tearing down pro-Barrow posters that read "Gambia has decided." It is unclear what the country's armed forces would do if the president refuses to resign and foreign troops arrive.
Barrow called on his countrymen to "work for a peaceful transfer of executive power, for the first time in our history since independence."
"If the colonialists could peacefully hand over executive power in accordance with the dictate of the people of The Gambia… [we] should be able to show a better example to our children," Barrow said.
He also spoke briefly about his plans for the future, pledging to keep The Gambia a free, secular and democratic country and announcing the creation of a think tank to outline a blueprint for the country's sustainable development. He also issued a call to others who might be able to contribute to crucial sectors to send their CVs to create a bank of experts.
Meanwhile, NPR reports that Omar Fay, The Gambia's ambassador to the US, has asked Jammeh to step down and is now being recalled. Local media report that a total of 11 ambassadors have made the same call, including the country's envoys in Beijing, Ankara, London Moscow, Brussels, Dakar, Madrid, Havana and Addis Ababa.