Medway NHS Foundation Trust said chairman Denise Harker and chief executive Mark Devlin had decided the time was right for them to move on.
The trust was criticised as failing and put in special measures following the Keogh Review in June.
It said a new chairman and chief executive were expected to be announced next month.
The Keogh Review found the trust, which runs Medway Maritime Hospital in Gillingham, had major failings.
It was criticised for poor communication with patients, poor management of deteriorating patients, inappropriate referrals, delayed discharges and long waits in A&E.
The hospital was later told by the Care Quality Commission it must take urgent action to improve safety on a maternity unit.
In November health regulator Monitor said the hospital's problems were broader than first thought and said it must take urgent action or face leadership changes.
The trust said in a statement Ms Harker and Mr Devlin had decided to hand over leadership of the trust to a new team.
"I would like to thank Mark and Denise for their dedication and integrity," said Colin Wilby, its senior independent director.
"Their commitment to public service is indisputable and we wish them both every success in the future."
Earlier this week it was announced Medway Maritime's A&E department would undergo a Â£5m redevelopment.
Mark Reckless, Conservative MP for Rochester and Strood, said the hospital had appointed a new medical director and a new chief nurse.
"I think they are doing good work and I have seen some improvements but frankly, given the problems, I am not sure its been fast enough," he said.

The full-back, 25, was hurt in Town's 3-1 defeat by Millwall on 9 April  - his first game since last December - and a scan has shown he needs surgery to repair ligaments.
"It's disappointing - he's just got back into the side," assistant boss Mike Jackson told BBC Radio Shropshire.
"Hopefully he'll get it sorted and be back playing as quickly as he can."
Demetriou, who is out of contract in the summer, has made 50 appearances for Shrewsbury, scoring three goals, since joining from Kidderminster Harriers in June 2014.
He spent the first half of this season on loan at Cambridge United and returned to the Shrews in January.
Following last season's automatic promotion, Shrewsbury are trying to avoid an immediate return to League Two over their final six games.
One point clear of the relegation zone, Town have a tough-looking start to their run-in with back-to-back home games against fourth-placed Bradford City and Sheffield United before trips to Gillingham, in sixth, and automatic-promotion chasers Walsall.
Micky Mellon's side then finish with two games against mid-table sides - Peterborough, currently 14th and Swindon in 16th.

Luis Suarez scored twice, while Neymar and Andres Iniesta also scored as Barcelona moved six points clear of Real at the top of the table.
BBC Sport reviews each player's performance, with BBC Radio 5 live co-commentator Chris Waddle, who was at the Bernabeu, giving his ratings out of 10.

The 22-year-old joined Nathan Jones' side in the summer after being spotted by chief recruitment officer Mick Harford playing for Truro City.
Vassell has scored twice in 23 appearances and the Hatters have taken up the option in his previous deal.
"Over the five months since he's been here he's done really well to impress us and grasp his chance," said Jones.

The 19-year-old follows Wayne Rooney, Raheem Sterling, Lionel Messi and Sergio Aguero in winning the award for Europe's best player aged under 21.
Rashford and Sanches' club team-mate Kingsley Coman also made the podium.
Portugal international Sanches will receive the award at a ceremony in Monte Carlo on Monday.
The prize, voted for by journalists across Europe, was established by Italian newspaper Tuttosport in 2003, who revealed this year's winner on their front page.
Sanches has played eight games for the German champions this season, having joined in the summer for £27.5m, and has 11 caps for Portugal.
At Euro 2016 - then aged 18 - he became the youngest player to feature in a European Championship-winning side and won the Young Player of the Tournament award.
Two bosses at a Kent hospital foundation trust put into special measures last year have resigned.
[NEXT_CONCEPT]
Shrewsbury Town defender Mickey Demetriou will miss the rest of the season because of a knee injury.
[NEXT_CONCEPT]
Barcelona took the honours in the first El Clasico of the La Liga season with a dominant 4-0 win at Real Madrid on Saturday.
[NEXT_CONCEPT]
Luton Town striker Isaac Vassell has signed a new contract with the League Two side until the summer of 2018.
[NEXT_CONCEPT]
Bayern Munich midfielder Renato Sanches has won the European Golden Boy award, beating Manchester United and England striker Marcus Rashford.