Adama Barrow announced the ceremony in a message posted on Facebook.
Last-ditch efforts by regional leaders to convince Yahya Jammeh to quit as president failed overnight. They have warned they could remove him by force.
Mr Jammeh, 51, lost elections last month, but wants the results annulled citing errors in the electoral process.
Mr Barrow, who is also 51, made the announcement on Thursday.
The message said he would be "sworn in today at 4pm (16:00 GMT)", inviting the general public to take part. He later said the ceremony would be "by invitation only" because the embassy had "a very limited capacity due to its size".
West African military forces, stationed at the border, are ready to enforce a transfer of power in The Gambia, a popular beach destination among European holidaymakers.
UN Security Council backing for intervention is being sought by Senegal and the regional bloc Ecowas, but some diplomats said if Mr Barrow requested help after his inauguration such approval would not be needed.
It is eerily quiet in The Gambia's capital. Most streets are deserted; shops, petrol stations and banks are all closed. People are mostly staying home uncertain about what may happen as European tourists continue to evacuate their hotels.
In some areas, men are standing on the roadside, arms crossed or looking at their phones. Some told us they were waiting for President Jammeh to go and would take to the streets once Mr Barrow was sworn in this afternoon.
They said they wanted West African troops to come in as soon as possible. Some also told us they were worried about Mr Jammeh's actions should there be an offensive against him. But so far we have seen little presence of security forces in the city.
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