In a statement on the club's website, Mr Mather said he has agreed to stand down in an attempt to help calm speculation over the governance and executive management of Rangers.
Mather said he was departing in order to try to restore calm to the club, two days after former oldco director Paul Murray won a courtroom victory to force a postponement of next week's schedule annual general meeting in a dispute over a possible election to determine whether he and his allies should join the board.
Mather replaced Charles Green as chief executive in April on an interim basis before taking on the role permanently.
Non-executive director Bryan Smart has also left, the club said in a statement to the Stock Exchange.
Mather said: "The interests of the club are of paramount importance and I believe these are best served by me leaving the club.
"Despite recent events and speculation, the facts of the matter are that the club is financially secure and in a far better place than it was a year ago.
"Unlike most football clubs Rangers has money in the bank, no borrowings and this season we have assembled a squad which is capable of progressing through the leagues.
"I have enjoyed a very constructive relationship with Ally McCoist and wish him and the team every success.
"My short tenure as chief executive has been beset by incessant attempts to destabilise the operations of the club, all done supposedly in the interests of Rangers.
"I had real faith in the rebuilding of Rangers and invested significantly in the club. Sadly, those who have been most active in upsetting the very good progress we have been making were not willing to do the same.
"I leave with my head held high and will remain as a shareholder and a supporter of Ally and his team.
"I would also like to pay tribute to the outstanding commitment and loyalty of Rangers supporters.
"No individual is more important than Rangers and my departure will hopefully alleviate some of the pressure surrounding the club and herald an end to the current hysteria, which I believe most fans desperately want to see.
"I have always tried to do my best for the club and the fans and I will continue my support of what is a fantastic club.
"There are a great many good and thoroughly decent people working with Rangers and I am proud to say that I was able to stand alongside them for a time.
"It is often forgotten that I put in £1m of my own money but I can assure everyone that it was never about the money for me.
"I consider it to have been my privilege and I am certain that once the Board is settled Rangers will be restored to the top of Scottish football.
"I wish Rangers and the fans every success in the weeks, months and years ahead. I will continue to follow the club's fortunes and support the team which is playing an exciting style of football. In fact, I hope to return to Ibrox and take in as many matches as my time will allow."
Rangers also confirmed to the London Stock Exchange the departure of non-executive director Bryan Smart.
Their statement read: "The company also announces that Bryan Smart has resigned from the board with immediate effect.
"The directors would like to thank both Craig and Bryan for their contributions to the club.
"As a matter of priority, the company, working with its advisers, will now seek to strengthen the board, both to meet the standards required for an AIM company, and to achieve stability through a consensus amongst major shareholders, which can then be considered by all shareholders at the annual general meeting."