The school will replace a Gaelic unit at Tollcross Primary, which is now full.
It will be built in Leith at Bonnington Primary, which is currently closed, and will undergo "substantial renovation" before re-opening in August 2013 as a stand-alone facility.
The Scottish Government is providing pound;1.8 million in capital funding, with the council funding the remainder of the pound;3.53 million project through prudential borrowing.
Minister for Gaelic Alasdair Allan said: "Edinburgh parents have clearly demonstrated their desire for a new Gaelic school in the city, with the existing facilities at Tollcross no longer suitable."
Tollcross Primary currently delivers Gaelic-medium education for 156 pupils in primary and a further 65 in nursery, but cannot take new pupils.
The council estimates there could be 300 pupils in the primary by 2017- 18.
Bord na Gaidhlig chair Arthur Cormack said: "We believe Gaelic schools are essential to growing the numbers of pupils in Gaelic education, offering additional opportunities to use the language in the whole school, enhancing their fluency in Gaelic."
There are two other Gaelic schools in Scotland - in Glasgow and Inverness.