Ms Norgrove, who was kidnapped in Kunar Province on 26 September, died on Friday as American troops attempted to rescue her. It was initially thought she had died at the hands of her captors.
Speaking at a press conference at Number 10 on Monday, Mr Cameron said new information has suggested Ms Norgrove could have died as a result of a grenade detonated by US troops during the rescue operation.
The PM said:
I want to assure Mr and Mrs Norgrove that I will do everything I possibly can to establish the full facts and give them certainty about how their daughter died.
My thoughts and the thoughts of the whole country are with them as they come to terms with the death of their daughter and this deeply distressing development.
Mr Cameron said that he believed the rescue operation should have gone ahead based on the advice that was given to the Government. He said Ms Norgrove’s life had been in “grave danger” from the moment she was kidnapped and it was believed that she would be passed up the terrorist chain and put at greater risk if a rescue was not attempted.
During the press conference, Mr Cameron also took questions on university funding and the Government Efficiency Review led by Philip Green.
On Monday evening, Mr Cameron spoke to US President Barack Obama, who offered his condolences for Ms Norgrove’s death.
A Downing Street spokesman said:
The Prime Minister and the President agreed that it was now essential to get to the bottom of what had happened in the course of the rescue operation. They looked forward to close co-operation between the UK and US authorities on the investigation and agreed to stay in close touch as it moved forward.
Foreign Secretary William Hague made a statement to the House of Commons about Ms Norgrove. Read the statement on the Foreign Office website.