In an internal email to staff, Brooks said she had written to Dowler's family to assure them that News International would "vigorously pursue the truth and that they will be the first to be informed of the outcome of our investigation".
She acknowledged the "devastating effect" of the practice on the family, but said she would not be resigning from her position.
"I am aware of the speculation about my position. Therefore it is important you all know that as chief executive, I am determined to lead the company to ensure we do the right thing and resolve these serious issues.
"We will face up to the mistakes and wrongdoing of the past and we will do our utmost to see that justice is done and those culpable will be punished."
Speaking on BBC Radio 4's the World at One programme this afternoon, News International director of corporate affairs Simon Greenberg echoed Brooks' statement.
"At the moment we don't know all of the facts ... But what I will say is that we are absolutely shocked and appalled and we are absolutely determined to get to the bottom of this."
Greenberg said he "understood" that people were calling for Brooks' resignation, but said "she has been absolutely clear that that is what she won't do ... She is determined to get to the bottom of this issue".
Discussing an internal investigation into the matter launched by Brooks, Greenberg said those working on it were "under no illusion that she is determined that if things went wrong, we will correct them".
Citing a News International executive, BBC business editor Robert Peston reported this morning that New Corp chairman Rupert Murdoch is backing Brooks "100 per cent".
Labour leader Ed Miliband this afternoon called on Brooks to "consider her position" at the company following yesterday's revelations about the Dowler case.
Calling for a public inquiry alongside shadow home secretary Yvette Cooper, Miliband described the episode a "stain" on British journalism and said Brooks should "consider her conscience and consider her position".
Miliband said earlier today that he was "shocked by the news of the hacking of Milly Dowler's phone".
"It beggars belief that anyone would undertake such a cruel and immoral act.
"The police inquiry must get to the bottom of who was responsible for this and who was complicit in it."
In a statement this morning, David Cameron said the claims were "quite, quite shocking".
"On the question of the really appalling allegations about the telephone
of Milly Dowler, if they are true, this is a truly dreadful act and a
truly dreadful situation.
"What I have read in the papers is quite, quite
shocking, that someone could do this actually knowing that the police
were trying to find this person and trying to find out what had
happened, and we all now know the tragedy that took place."
The Dowler family lawyer Mark Lewis announced yesterday that Milly Dowler's parents, Sally and Robert, would be taking legal action against the News of the World.
According to Lewis, their phones were also targeted
by the tabloid and they were made aware of the hacking by the Met police
in April, a month before Bellfield went on trial.
The evidence is understood to
have come from a cache of 11,000 pages seized by the Met Police's Operation Weeting team from private investigator
Glenn Mulcaire when he was arrested in 2006.
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