NEW YORK – The stock market staged a huge rally Thursday after investors got the aggressive economic help they wanted from the Federal Reserve.

The Dow Jones industrial average spiked more than 200 points and cleared 13,500 for the first time since the beginning of the Great Recession. The average is within 625 points of its all-time high.

The Fed said it would buy $40 billion of mortgage securities a month until the economy improves. It left open the possibilities of buying other assets and of buying long after the recovery picks up.

The central bank also extended its pledge of super-low short-term interest rates into 2015, and extended a program to drive down long-term rates.

It was the package known as QE3 — a third round of quantitative easing, in market-speak. And it was just what investors were hoping for.

“They’re saying that the punch bowl, the fuel for the economy, isn’t going away — it’s going to be here as long as you need it,” said Tony Fratto, a former aide to President George W. Bush and managing partner at Hamilton Place Strategies, a policy consulting firm in Washington.

The Dow closed up 206.51 points, the seventh-biggest gain this year, at 13,539.86, its highest close since the last days of December 2007, the first month of the recession.

The broader Standard & Poor’s 500 index was up 23.43 points at 1,459.99, also its highest since December 2007. The Nasdaq composite index, which has been trading at its highest levels since 2000, was up 41.52 at 3,155.83.