Consumers spent more, especially on health care and utilities, and businesses invested more, in software and vehicles among other items, spurring the fastest growth in a year. The nation’s total output of goods and services grew at an annual rate of 2.5 percent from July to September, almost double the 1.3 percent rate in the previous quarter, the Commerce Department estimated on Thursday.

That pace is not brisk enough, however, to recover the ground lost in the economic bust, relieve unemployment or even entirely dispel fears of a second recession.

“It ain’t brilliant, but at least it’s heading in the right direction,” said Ian Shepherdson, the chief United States economist for High Frequency Economics, a data analysis firm. “I want to see 4 percent, but given that people were talking about a new recession, I’ll take 2.5 or 3.”

Investors embraced the domestic report and a broad agreement struck by European leaders to resolve their debt crisis, causing some major stock indexes to soar by 5 and 6 percent in Europe and by around 3 percent in the United States, where the Dow Jones industrial average closed above 12,200.

Still, one did not have to look far to find cautionary signs in the American economic report. Economists do not expect growth to accelerate in the next few quarters to the point that it drives the unemployment rate well below 9 percent. The improvement is simply not enough to be perceptible to anxious American families.

“For most people, they’re unable to really make a distinction between a recession and just 2 percent growth, which means the economy is growing so weakly it can’t hire enough people to make a dent in unemployment,” said Bernard Baumohl, the chief economist at the Economic Outlook Group.

A 2.5 percent growth rate may be hard to sustain, noted Kathy Bostjancic, director for macroeconomic analysis at the Conference Board, which tracks consumer and executive sentiment. Real income is declining, housing prices are stalled and, as the National Association of Realtors reported on Thursday, home sales in September were down for the third consecutive month. Personal disposable income, adjusted for inflation, fell 1.7 percent in the third quarter, its biggest drop since the third quarter of 2009.

While income was falling, consumer spending rose at an annual rate of 2.4 percent, more than triple the rate in the second quarter.

So where did the money come from? Consumers put away less in savings, and credit card debt inched upward.

“That is unlikely to continue if the economy grows weakly because Americans are much more conscious about adding on a lot of debt to their balance sheet,” said Ms. Bostjancic, who added that the negative outlook had begun to spread to businesses.

“C.E.O. confidence is starting to melt away, along with consumer confidence levels, which have always been low,” she said.

Car sales were strong in July, August and September, and many analysts pointed to those numbers as evidence that consumers were confident enough to take out loans. But the seasonally adjusted number in Thursday’s report showed a slight decline in consumer spending on cars, leading economists to speculate that businesses were responsible for the increase in sales.

Most of the increase in consumer spending was instead driven by nondiscretionary expenses like health care, according to an analysis by David A. Rosenberg, chief economist for Gluskin Sheff, a wealth management firm. “We can understand why consumer confidence has sunk,” he wrote, “when spending on essentials such as utilities and medical bills have to be funded by drawing down the personal savings rate.”

Declines in the savings rate as large as the one reflected in the report — one percentage point, to 4.1 percent of income — are rare, and more than half the time, they herald recession, Mr. Rosenberg said.

Businesses increased their capital investments at a 17.4 percent annual rate. Business spending has been strong throughout the recession, a hopeful sign because investment in factories, offices, equipment and software often precedes hiring.

Some economists also saw a bright spot in inventories, which grew less than expected in the third quarter. That most likely means there are few excess goods piled up, and businesses may need to replenish their inventories in the fourth quarter, producing stronger growth.

This article has been revised to reflect the following correction:

Correction: October 27, 2011

An earlier version of this article gave an incorrect name of the company where Nigel Gault works.  It is IHS Global Insight, not HIS Global Insight. 

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