The move reversed decisions made by longtime former president Yahya Jammeh.
The visit will be his first to the continent as Britain's top diplomat. "I'm delighted to be the first foreign secretary to visit Gambia this week", Johnson said in a statement Monday.
"As one of his first acts in office, President Barrow's notification to the United Nations secretary-general of The Gambia's decision to reverse withdrawal from the ICC is a crucial victory for victims of grave crimes and the rule of law", Cl?ment Capo-Chichi, Africa regional coordinator with the Coalition for the International Criminal Court.
Barrow - a former businessman who lived in London for three years - is determined to bring Gambia's development to the level its citizens expect him to. "Their elections highlight the continuing strengthening of democracy in West Africa", Johnson said ahead of the trip.
Mr Jammeh took The Gambia out of the Commonwealth in 2013, calling it a neo-colonial institution.
Jammeh, who mockingly called the ICC the "International Caucasian Court", flew into exile last month under global pressure after losing to Barrow in the December election.
He characterised the Commonwealth as having "crowds of flag-waving piccaninnies", using a derogatory term for black people that caused outrage.
Actual withdrawal from the ICC comes a year after notification.
Diplomatic sources have said Britain is likely to target justice reform as an area in which it can provide expertise to the new government.
Tourists were flown out of the country en masse in January after Jammeh declared a state of emergency when he lost the election to Barrow but refused to stand down.
Mr Johnson and President Barrow will also hold discussions on fighting extremism and terrorism and how to support the growing tourism industry in Gambia.