The Dow Jones industrial average rose nearly 332 points to 11,615.77. Gainers on the New York Stock Exchange outnumbered losers by a 3-to-1 margin.
The Nasdaq Composite Index gained 64 points at 2,349.83. And the Standard-and-Poor's 500 Stock Index was up 36 points at 1,284.88.
Trading volume on the New York Stock Exchange totaled 5.2 billion shares. Volume on the Nasdaq Market came to 2.3 billion shares.
Investors were fired up after the Federal Reserve left interest rates unchanged and assuaged some of the market's fears about the economy.
In a statement accompanying its widely expected rate decision, the Fed said "economic activity expanded in the second quarter, partly reflecting growth in consumer spending and exports." That assessment was welcome news to a market that has feared the economy was falling into recession because of weak consumer spending.
Stocks were already moving sharply higher before the Fed meeting in response to a report that services-sector activity fell less than expected last month and another big drop in oil prices that took crude as low as $118 a barrel.