Robert Hannigan, currently the director of defence and intelligence at the Foreign Office, will replace Iain Lobban, who was in the post for six years.
The Snowden leaks showed Government Communications Headquarters - a giant doughnut-shaped building in Cheltenham in southwest England - was a key player in covert US surveillance operations worldwide.
Foreign Secretary William Hague said he was "delighted" by the appointment, and that Hannigan "brings to the job a wealth of relevant experience in the fields of national security, counter-terrorism and international relations."
Hannigan was former PM Tony Blair's chief advisor during the Northern Ireland peace process, and was the prime ministerial security adviser from 2007 until 2010, when he took up his current position at the Foreign Office.
Hannigan said it was a "privilege to be asked to lead GCHQ, an organisation which is so central to keeping the people of this country safe" and said he was "excited about meeting the challenges of the coming years".
"I have great respect for the integrity and professionalism of the staff of GCHQ and for what they have achieved under the outstanding leadership of Iain Lobban," he said.
His challenges will include shoring up GCHQ's reputation after the Snowden revelations.
One of the leaks revealed that Britain was running a secret Internet monitoring station in the Middle East, intercepting phone calls and online traffic, with the information processed and passed to GCHQ and then accessible by the United States.
According to Snowden, it has also tapped into more than 200 fibre-optic telecommunications cables, some of them transatlantic, and was handling 600 million "telephone events" each day.
When called to appear before a parliamentary committee last November, Lobban insisted the agency was not conducting mass snooping on the British public.
"We do not spend our time listening to the telephone calls or reading the e-mails of the majority," he said.It was the first time a head of GCHQ had given evidence in public.