White's appointment was announced to the stock exchange by McAlpine on Wednesday morning, ending intense market speculation over whether O'Rourke, the chief executive of Laing, was about to appoint someone to take over day-to-day operations.
The announcement came after Building telephoned White to enquire if he had been asked to take the post.
White, 48, whose career started at Laing in the 1970s, is expected to be made chief operating officer at the company in April.
Oliver Whitehead, chief executive of Alfred McAlpine, said: "We decided to come clean.
There have been rumours within the company for a few weeks about Andy, but it has now come out in the industry. I am disappointed to see Andy go, but you have to be grown up about this.
"I think he will find it a different company to the one he left. Ray [O'Rourke] wants to recreate a good business."
O’Rourke needed a decent name to head it, and White is that. It’s a big-ticket job for him
Stephen Rawlinson, analyst, Peel Hunt
Analysts claimed that White's departure was a blow for McAlpine. Its share price fell 8.5p to 446.5p soon after the announcement.
Peel Hunt analyst Stephen Rawlinson said: "O'Rourke needed a decent name to head Laing, and White is that. It's a big-ticket job for him."
Analysts said O'Rourke may tap into White's experience at McAlpine's special projects arm.
The operation, which has a record of making 5%-plus margins, finds and acquires land for developers and takes a stake in special purpose vehicles to develop schemes on the sites. White's experience could enable O'Rourke and Laing to extend their range of services in this area.
White will be replaced by the return of Matt Swan, who left McAlpine last year with former McAlpine Homes boss Graeme McCallum to join housebuilder Wilson Connolly.
O'Rourke declined to comment on White's new role, but industry sources said his resignation could lead to more departures and appointments. One insider said: "This could spark redundancies. Any new boss wants to make his mark on a firm when he takes over."