His new role will be as the inaugural chief executive of the Central Adelaide Local Health Network, which came into being this month.
When Mr Rousseau's resignation was announced in June he advised he would stay until the end of the year and hoped to help with the transition to the new appointee. However, the board has approved his request to finish on September 16.
Mr Rousseau said he was disappointed he would not be getting a break between jobs. He takes up his new role on September 19.
While the Southern District Health Board is still trying to work out how it can fund the upgrading of its hospital facilities, Mr Rousseau said Royal Adelaide Hospital's rebuild was already approved, although "they haven't turned any soil yet".
Mr Rousseau, whose salary here is between $490,000 and $500,000, said he would prefer not to discuss the salary for his new job.
He described the new job as exciting, although one of the down sides was that he would not be able to help the transition to the new chief executive, who may not arrive until next year.
Advertisements for the job should begin appearing this weekend.
Asked if he had been head-hunted for the job, Mr Rousseau said the position had arisen immediately after his resignation and he had applied for it.
He will oversee more than 8700 staff , compared with about 4500 in his existing role.
The 650-bed hospital is the main tertiary hospital in Adelaide, but also treats patients from throughout South Australia.
Its emergency department deals with more than 63,000 patients a year.
Dunedin Hospital's ED treats about 36,000 people a year.
The Southern District Health Board is expected to discuss at its August meeting how the chief executive's position will be covered in the time between Mr Rousseau's departure and the new appointment.
