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The Fed is expected to issue a statement at 2 p.m. EDT after a two-day meeting. It is widely expected to continue winding down its quantitative easing bond purchases by another $10 billion to $45 billion and make slight tweaks to its statement.
"This may color how they talk about the economy. It may not be as optimistic as some expected," Swonk said. But she thinks the Fed will go on with its plans to taper quantitative easing.
Stocks were subdued, initially weaker on the GDP report, but higher in late morning trading. Treasurys were higher after the report, and the 10-year yield, which moves inversely, fell to 2.66 percent.
Read MoreYikes! Growth slowed to a crawl in first quarter
"The fact that you've not heard anyone say the word 'recession' and equity futures were flat tells you all you need to know," said Daniel Greenhaus, chief global strategist at BTIG. "This is being written off appropriately as not a qood quarter but it's largely weather impact."
The GDP report gave a number of clues about the weather impact. Consumer spending on goods rose only 0.4 percent as cold weather slowed shopping trips, shut businesses and closed schools, but total consumer spending rose at 3 percent. That was largely due to spending on energy to heat homes and on health care, possibly as more consumers signed up under the Affordable Care Act.
J.P. Morgan economists already knocked down their forecast for first quarter growth to zero, following government revisions Wednesday to retail sales data for the first quarter. The 3 percent increase in consumer spending, therefore, slipped to a 2.9 percent pace, they wrote in a note.
"It looks like the economy hit an air pocket in Q1. It stalled out. The share of the weak growth can be attributable to the temporary weights on the economy. Weather probably cut 1 point from growth. The expiration of emergency unemployment insurance cut as much as a half point, and of course the inventories subtracted just over a half point. If you add that up, the economy grew 2 percentage points in Q1, but that's still weak," said Mark Zandi, chief economist at Moody's Analytics.
The economy was growing at a pace of 3.4 percent in the second half of last year, and many economists expect the current quarter growth to show the spring back effects after record snowfalls and cold dampened spending and business activity. Zandi said the weak number does not change his expectation for 3.5 percent second quarter growth.
"I'm thinking we could see big revisions to the initial numbers. There's a lot of assumptions in here. At the end of the day, I look at it and say, this will give the Fed more pause today. They'll have to describe it as the weather impact was bigger than expected and they'll to say we think it's transitory and we'll wait for more data," Swonk said.
The change in private inventories subtracted 0.6 percent from growth, potentially a positive for the second quarter as businesses increase output to meet demand.
Business spending fell at a 2.1 percent pace in the first quarter, the first decline in a year and a disappointment after the fourth quarter's 5.7 percent gain. Exports fell at a 7.6 percent pace in the first quarter.
The Fed will have studied the data ahead of drafting its statement. "I'm sure they're surprised, too. It's at least a point less than general consensus. The inventory accumulation was less but it's still very high so there's probably still more weakness related to weather in Q2," said Zandi. "This would steer them to be more dovish. I don't think their forecast will change because of this. Mine hasn't, but I'm more on high alert."
Encouraging to economists Wednesday was the better-than- expected ADP report, looked at as a sort of barometer for the April jobs report. Economists expect about 210,000 nonfarm payrolls for April, when the government jobs report is released Friday.
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