The presenter, 62, has been off-air since last July after it was alleged the images started when the person was 17.
Edwards, whose earnings were £435,000 to £439,999 last year, was the broadcaster's highest-profile news host, often chosen to present its coverage of major events.
He revealed the death of Queen Elizabeth II to the nation in September 2022.
The BBC yesterday said: "Huw Edwards has resigned and left the BBC.
"After 40 years of service, Huw has explained that his decision was made on the basis of medical advice from his doctors.
"The BBC has accepted his resignation which it believes will allow all parties to move forward.
"We don't believe it appropriate to comment further." The corporation said Edwards had not been awarded a payoff as part of his departure.
A source within the BBC told the Daily Express: "No he has not. Neither has he received a payment in lieu of his notice period.
"He has resigned and leaves with immediate effect."
It was not known last night if the BBC's internal workplace investigation into whether Edwards had brought the broadcaster into disrepute had been completed before he stepped down.
His exit comes nine months after his wife Vicky Flind had revealed that Edwards had been admitted to hospital with "serious mental health issues".
Police did not take action against him, saying that there was no evidence that a criminal offence had been committed.
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