From 2012-13, students from the rest of the UK will pay ?9,000 a year at Edinburgh, the same level as at many English universities. However, because Scottish degrees last four years, the total cost to students will be up to ?36,000.
Scottish and European Union students who study in Scotland will continue to be exempt from fees.
Forty-three per cent of Edinburgh’s full- and part-time first-degree students in 2009-10 were from the rest of the UK.
The university said the fees were being brought in because the Scottish government had removed central funding for non-Scottish students.
In June, the Scottish education secretary, Michael Russell, announced that universities north of the border would be allowed to charge rest-of-UK students up to ?9,000 a year so that Scottish students were not “squeezed out” by other applicants taking advantage of much lower fees.
Both the University of Aberdeen and Heriot-Watt University have announced a maximum fee of ?,000 for rest-of-UK students over their four-year degrees.
Mr Russell has predicted an average annual fee of ?6,375 for rest-of-UK students at Scottish institutions.
Rest-of-UK students from the most economically disadvantaged backgrounds who study for four-year degrees at Edinburgh will be eligible for bursaries of up to ?28,000 and access awards of up to ?5,000, the university said.
Graeme Kirkpatrick, deputy president of NUS Scotland, said that a ?36,000 degree was “staggering and ridiculous”.
"The average cost to study at Oxford and Cambridge is around ?25,000 in fees, which while still eye-wateringly large, pales in comparison with this," he said.
He added: “Given that students from the rest of the UK make up about 15 per cent of the whole of Scotland’s university sector, they are clearly taking huge risks with tens of thousands of people’s futures and many millions of pounds.”