The Metropolitan Police say no criminal offence has been committed by Huw Edwards, after the presenter was named as the BBC star facing allegations over payments for sexually explicit images.
Edward's wife Vicky Flind named her husband as the presenter facing the allegations on Wednesday evening.
Meanwhile, detectives from the Met's Specialist Crime Command concluded their assessment and determined there is no information to indicate that a criminal offence has been committed.
Sources have also made it clear to the PA news agency that Edwards has not resigned.
A BBC spokesperson said: "We have seen the statement from the police confirming they have completed their assessment and are not taking further action. We're grateful to them for completing this work at speed.
"The police had previously asked us to pause our fact-finding investigations and we will now move forward with that work, ensuring due process and a thorough assessment of the facts, whilst continuing to be mindful of our duty of care to all involved."
Vicky Flind released a statement on her husband's behalf on Wednesday evening.
Ms Flind said: "In light of the recent reporting regarding the 'BBC Presenter' I am making this statement on behalf of my husband Huw Edwards, after what have been five extremely difficult days for our family.
"I am doing this primarily out of concern for his mental well-being and to protect our children."
The statement added: "Huw is suffering from serious mental health issues. As is well documented, he has been treated for severe depression in recent years.
"The events of the last few days have greatly worsened matters, he has suffered another serious episode and is now receiving in-patient hospital care where he'll stay for the foreseeable future.
"Once well enough to do so, he intends to respond to the stories that have been published. To be clear Huw was first told that there were allegations being made against him last Thursday.
"In the circumstances and given Huw's condition I would like to ask that the privacy of my family and everyone else caught up in these upsetting events is respected. I know that Huw is deeply sorry that so many colleagues have been impacted by the recent media speculation. We hope this statement will bring that to an end."
South Wales Police said in a statement: "South Wales Police has remained in contact with representatives of the Metropolitan Police and the BBC following a meeting on Monday.
"Information was initially received by the force in April 2023 regarding the welfare of an adult. No criminality was identified.
"Following recent events, further enquiries have been carried out and officers have spoken to a number of parties to establish whether any criminal allegations are being made.
"At this time, there is no evidence that any criminal offences have been committed. There are no ongoing enquiries being carried out by South Wales Police.
"However, should evidence of criminality or safeguarding issues be identified at any point in future then they will be investigated."
Don't miss the latest news from around Scotland and beyond - Sign up to our daily newsletter here .
