Legally, it needs at least 5% of local electors for a referendum to go ahead.
Those behind the petition say a mayor would offer more dynamic leadership and "rise above party politics".
Council leader Tim Warren said it felt "unnecessary" so soon after the elections. If successful, a directly elected mayor would replace him.
The council announced on Monday that it had received 6,818 valid signatures. A further 2,789 signatures were ruled to be invalid.
Directly elected mayors were created by the Local Government Act 2000 as one option for local government, as long as the idea was backed in a referendum.
Bath & North East Somerset Council said the threshold set out under the 2000 Act had been reached and "it is anticipated that the referendum will be held early next year".
In Bath, the council is run by a leader and cabinet system.
Philip Raby, one of the team working for a referendum, said they favoured holding it on 5 May 2016, the day of the police and crime commissioner elections.
He felt a directly elected mayor would offer "visible and accountable leadership" for the wider area - something he felt the current system did not offer.
Mr Warren said there were "challenging budgetary issues" to concentrate on and questioned whether there was much public appetite for a referendum.
Local petitions sparked referendums which resulted in elected mayors in Salford, Greater Manchester in 2012 and in Copeland, Cumbria in 2015.
The Localism Act 2011 allowed central government to trigger referendums for elected mayors - but of the ten held in May 2012, only Bristol voted in favour of getting a mayor.

The sturgeon, named Steve, swam out of World of Water in Romsey, Hampshire when it was inundated by flood water in early February.
But now staff at a car wash a mile from the centre have discovered him in a deep puddle.
A larger koi carp fish named Chadwick escaped at the same time but, despite sightings, has not yet been retrieved.
Centre manager Mark Bradbury said he was "hugely relieved" Steve had been recovered and "only a little worse for wear for his travels."
The seven-year-old fish was used as part of the shop's displays, but Mr Bradbury said similar fish would retail for up to Â£1,000.
"He must have crossed the road and got to the petrol station through the flood water. He's got a few bumps and scrapes but is happy to be back with his friends," he explained.
The centre has now re-opened after being closed for three weeks because of the floods.

The Addicks dropped into the relegation zone following Tuesday's 3-0 defeat at home to Preston.
The 40-year-old has been in charge at The Valley since January.
"I need to worry only about the next game, how I pick the team and how I encourage the squad to bring the result on Saturday," he told BBC Radio London.
Charlton have only won two of their 12 league games this term and Luzon described the defeat by Preston as "a difficult night".
However the Israeli, who previously worked under Charlton owner Roland Duchatelet at Belgian side Standard Liege, says recent injury problems have hampered his team selections.
"We know that we have a crisis with the injuries in the last two months," Luzon said.
"This is the situation. This is our squad. Without a pure wide player in the eleven, it is difficult to create chances and get the result.
"I hope on Saturday some of them will come back and help us to choose a normal eleven with balance. Then we can play football."
A referendum on an elected mayor for Bath and North East Somerset will be held in 2016, after a petition reached the threshold of 6,437 votes.
[NEXT_CONCEPT]
A metre-long fish that escaped from an aquatic centre during recent flooding has been found.
[NEXT_CONCEPT]
Charlton Athletic head coach Guy Luzon is not concerned about his future despite a run of eight games without a win in the Championship.