Murdoch, who is visiting the group's London headquarters in Wapping today, has sent an internal email to staff announcing the imminent launch.
He wrote: "We will build on the Sun's proud heritage by launching the Sun on Sunday very soon.
"Our duty is to expand one of the world's most widely read newspapers and reach even more people than ever before. Having a winning paper is the best answer to our critics.
"I am even more determined to see The Sun continue to fight for its readers and its beliefs. I am staying with you all, in London, for the next several weeks to give you my unwavering support. I am confident we will get through this together and emerge stronger."
Journalism.co.uk learned last month that News International was planning to launch the Sunday version of the popular tabloid newspaper on 29 April.
The date was then brought forward, but doubts about the launch were raised following the latest series of arrests, in which nine current and former Sun journalists and executives were arrested and bailed.
Staff have been secretly working on the new publication since January and it is believed some former News of the World employees (casual and/or full-time) are involved.
Murdoch also announced that the suspensions of all staff who were recently arrested over alleged corrupt payments to police and other public officials will be lifted "and they are welcome to return to work".
Five senior journalists at the Sun were among eight people arrested in dawn raids on Saturday. None of them have been charged and they are on police bail.
They were named in the press as deputy editor Geoff Webster, picture editor John Edwards, chief reporter John Kay, chief foreign correspondent Nick Parker and a news editor, John Sturgis.
Murdoch said News Corporation would "obey the law" and continue to co-operate with the police investigations.
He added: "We will continue to ensure that all appropriate steps are taken to protect legitimate journalistic privilege and sources, which I know are essential for all of you to do your jobs. But we cannot protect people who have paid public officials."
Murdoch's full email to staff has been published on our editors' blog.