Mark Bailey has also stepped back from his ministry as Team Rector of Trinity Cheltenham.
In a statement on the New Wine website, the trustees said it was with the "greatest sadness and regret" that they had to accept his resignation.
They're calling on Christians to pray for him, his family and his church.
His resignation comes following a meeting, instigated by Mr Bailey, with the Bishop of Gloucester, Rt Revd Rachel Treweek.
Following the meeting, Bishop Rachel issued a statement to members of Trinity Cheltenham. She said: "While I can't go into any detail, I can tell you that this relates to a matter now being considered under the Clergy Discipline Measure.
"This is not a matter involving the police or other statutory agencies. Further information will be made available at the appropriate time."
In the New Wine Trustees statement, they said: "[We] recognise and share the deep distress that this news will cause. [We] are profoundly conscious of their need for God's guidance and wisdom in the weeks ahead and encourage all friends of New Wine to pray for them, for New Wine itself and for all those personally involved."
Former leader of New Wine John Coles has been appointed interim chair of the trustees.