The board's unanimous decision represents a U-turn for chairman Jim Scrimshaw who said last week that even if only a minority of colleges lacked confidence in them, they would resign.
The decision was made at a board meeting as The TES went to press. Yesterday the board was due to convene an open meeting in Birmingham for chairs and principals. It was to be held to allow questioning about their handling of the departure of former chief executive Roger Ward, and the inquiry set up to examine allegations of impropriety against him.
More than 150 signatures have been collected in support of an emergency general meeting to consider sacking the board but many principals preferred to avoid that option and hoped they would stand down - even if they stood later for re-election. Last week Mr Scrimshaw agreed that an EGM would be damaging to the sector.
Mr Scrimshaw said the board had to stay to safeguard the continuity and strength of the association as the voice of the sector.
"There is an established democratic procedure, agreed by all member colleges, whereby members who were dissatisfied with the performance of the board can call for their removal through a motion which must be carried at an EGM.
"We believe we have widespread support for the actions we have taken in the past two months I To stand down over this issue would set a wholly unwise precedent. No board can be expected to work to the complete satisfaction of every college at any one time."
Colin Flint, principal of Solihull, who was calling for the board's resignation, said: "This will be a major disappointment to very many members who believe the correct and honourable decision is for them to stand down. There is a real danger that some colleges will not continue in membership if this position continues to be maintained."
The council of the Association of Principals of Colleges urged the board to remain in office. Hilary Cowell, on the AOC board, is president of the APC.