But he still faces an investigation by the Department for Education and Employment and could be banned from teaching for life.
The Government inquiry comes despite last month's Crown Court decision that Mr Edson, 32, was not guilty of sexually assaulting Rachel Russell, now aged 16.
The couple began their relationship when Rachel went to his house to babysit his daughter.
Last week the Government confirmed it is to press ahead with new laws banning teachers from conducting affairs with 16 and 17-year-old pupils. An "abusive relationships" amendment to the sexual offence Act is expected in the next parliamentary session and could, it is estimated, result in the prosecution of 10 to 15 teachers each year.
The teaching unions are worried about the proposals, arguing they could criminalise relationships with any 16 or 17-year-old, even those who are not pupils.
But Home Office Minister Lord Bassam insisted the laws are necessary. "While some teachers may be concerned about bringing the criminal law into this field I think there is broad acceptance that we have to set standards and that is in effect what the law is trying to achieve here," he said.
Lord Bassam denied that the proposed law would be a "sledgehammer to crack a nut".
A DFEE spokesman said Mr Edson's resignation would automatically trigger an inquiry into his conduct: "We will investigate cases regardless of whether the teacher is convicted or not or if they have resigned. We would then consider whether to bar them from teaching."
Mr Edson's headteacher, John Fryett said: "Earlier this term the school asked Mr Edson to consider resignation in view of his public admission of an inappropriate relationship with a student at the school in 1998.
"The governors and the headteacher, on behalf of the school, wish to make clear their disapproval of Mr Edson's actions and their regret that he chose to act this way."
Last week the Home Office issued guidance to schools, youth groups and other organisations on how to avoid inappropriate relationships and sexual abuse.
These have been welcomed. "At long last, principles have been mapped out which clarify and protect young people, teachers and all who work in the caring services," said a spokeswoman for the National Union of Teachers.
Guidelines will assist adults by specifying boundaries of behaviour, according to Michael Marland, who is head of North Westminster community school, London, and involved in the National Association of Pastoral Care.
He said: "You have to be extremely careful not to put a youngster in an embarrassing situation, even if your intentions are extremely good. Guidelines would be helpful and make life easier but it is important that we don't generate a climate of fear."
AFFAIRS IN THE CLASSROOM
What the Government says should happen now if a sexual relationship is suspected between a teacher and a student over 16.
Staff should report allegations of relationship to the headteacher.
Headteachers should make initial inquiries and, if allegation has substance, involve social services, making sure disciplinary and child protection procedures are kept apart.
Head should consult education authority before deciding to suspend the teacher. Suspension is justified if: the child is at risk; dismissal for gross misconduct is possible; or suspension would help the investigation of the complaint.
Chair of governors and chief education officer should be informed in writing of suspension.
Governors should only be given minimum information so they are not prejudiced should they be involved in any subsequent disciplinary or appeal proceedings.
Suspended teacher must be given name of council contact to keep them informed of progress.
If allegations are justified, school can dismiss the teacher, subject to usual disciplinary processes.
Schools must report allegations to the Department for Education and Employment if the teacher resigns or is dismissed. DFEE will carry out its own investigation before deciding whether to ban him or her from teaching.