The Cyrenians charity is shutting its operation in St George's West Church, Shandwick Place, after seeing its funding cut from more than 30,000 to 7627 by the city council.
The centre is used by people moving to the Capital from abroad – ranging from eastern European economic migrants to those fleeing persecution and war in their home countries.
The centre was set up in 2004 to help people find homes, health services, schools and work when they first arrived in Edinburgh.
Although Edinburgh's asylum seeking and refugee populations are small compared with many other, similar sized UK cities, supporters say the centre was extremely well-used.
It has helped many Poles and other Europeans who have arrived in Edinburgh with little or no English.
Manager Neil McCulloch said the centre had been "fantastically effective" and the charity would be working to ensure the needs of its users would be taken care of in the future.
However, Nina Giles, director of Edinburgh and Lothians Racial Equality Council, said it would be a "big loss".
"It is not just a resource for refugees but also new arrivals from eastern Europe," she said.
"It is a good source of information on housing, immigration and other services that people require."
The number of asylum seekers and refugees living in Edinburgh has been estimated at 1200, while around 35,000 Poles have settled in the Lothians – the highest in any part of Scotland. The city council defended its decision to cut funding to the Cyrenians to run the centre.
Councillor Paul Edie, the city's health and social care leader, said: "The decision was taken to reduce the funding awarded to the Edinburgh Refugee Centre by the council in March. The number of asylum seekers and refugees in Edinburgh has greatly decreased in recent years and there is therefore a much-reduced need for the kind of services offered at the centre.
"There are also some other sources of help available. Although the decision to cut grants was a difficult one, we have to make sure that the taxpayer gets best value for money. Our aim is always to balance the books while maintaining and improving key frontline services."
Councillor Ewan Aitken, leader of the city's Labour group, criticised the decision.
"In a world of huge global migration we are shutting down the refugee centre," he said. "We spend a fortune on tourism so the message seems to be 'Come to Edinburgh, unless you're a refugee'."