Associated Newspapers owner Lord Rothermere has apologised in writing to Labour leader Ed Miliband for the conduct of journalists on the Mail on Sunday.
But he has not yet become drawn into the row over the Daily Mail's description of Miliband's late father Ralph as "the man who hated Britain".
Mail on Sunday editor Geordie Greig has apologised and suspended two journalists after a reporter approached a member of Miliband's family at a memorial service for his uncle Harry on Wednesday.
A Labour spokesman said: "Lord Rothermere has repeated the apology for the behaviour of the Mail on Sunday. This is an important step.
"However, he says he does not believe it reflected the culture and practices of the Mail or Mail on Sunday, and also he does not address the treatment of Ed Miliband's dad over the last few days.
"We continue to believe these issues need addressing and until they do so, many people will continue to believe that these newspapers are not upholding the values and decency of the British people."
The Daily Mail continues to stand by its report of Saturday and its subsequent suggestions that Miliband's attacks are politically motivated – coming as they do ahead of a crucial Government decision next week on the future of press regulation.
The Mail today carried a page four lead story about the Mail on Sunday apology and noted: "The Daily Mail and the Mail in Sunday are automomous papers with their own editors and separate staff."
Daily Mail editor Paul Dacre is also editor-in-chief of the Mail on Sunday but evidently has no involvement in the paper editorially on a day to day basis.
Conservative MP and chairman of the Commons Culture Committee John Whittingdale said the Mail piece about Ralph Miliband was "somewhat offensive" but he said: "The Daily Mail has a very robust stance and that is part of a healthy press".
Executive director of the Society of Editors Bob Satchwell told the Mail: "What we are dealing with is essentially a political row between two camps..
"An editor should be free to edit the papers…the readers are their judges."