Jeremy Fleming, currently deputy director-general of the domestic spy agency MI5, will take up his new post at Government Communications Headquarters next month.
Fleming "is a dedicated public servant whose work over two decades in the intelligence services has helped to keep our country safe", Foreign Secretary Boris Johnson said in a statement.
Trump repeats charge that Obama ordered wiretaps against him
He is taking over from Robert Hannigan, who announced in January that he was stepping down for personal reasons.
This month, a Fox News commentator, Andrew Napolitano, aired claims that president Barack Obama had used GCHQ to wiretap president-elect Trump as a way of circumventing laws on spying on US citizens.
The claims were then repeated by White House spokesman Sean Spicer at a briefing on Thursday.
FBI director to testify on Russia ties, alleged wiretap
In a rare rebuke, a GCHQ spokesman swiftly denounced the allegations as "nonsense", saying they were "utterly ridiculous and should be ignored".
Prime Minister Theresa May's spokesman repeated the denial on Friday and said British officials had spoken to the White House and received assurances that the claims "will not be repeated".
Trump on Friday reiterated his charge that Obama had ordered a wiretap against him but said he did not endorse the Fox News claim that GCHQ carried it out.
"We said nothing" about the GCHQ claim, Trump told journalists.
"That was a statement made by a very talented lawyer on Fox. And so you shouldn't be talking to me, you should be talking to Fox," he said.
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