Mr. Duncan called all 15 governors on March 29 to inform them of the two winners.

“I didn’t know how those calls would go,” Mr. Duncan said. “You know, you never want to call folks with bad news. And I couldn’t have been more impressed with their commitment and their desire to take the next step in Round 2.”

Joanne Weiss, an aide to Mr. Duncan who is administering the competition, said she and her staff were working hard to persuade states to improve their proposals for the second round. “But it’ll be up to them, of course, to decide whether it’s worth their time and resources,” Ms. Weiss said.

A new rule capping award money, which is to be spent over four years, is causing states to waver. California, which requested $1 billion, can now only hope to win $700 million. Louisiana, which asked for $314 million, is now capped at $175 million, as is Colorado.

South Carolina, which hoped to win $300 million, came in sixth and is now capped at $175 million.

“That’s a lot of money, and we need it,” said the state’s superintendent of education, Jim Rex. “But spread it over four years, with all the federal expectations that come with it, and you have to ask whether you have the time and capacity to gear up again for the arduous work of filing a new proposal. We’re still weighing that.”

Florida has already decided to reapply, said Eric Smith, the state’s education commissioner. But because the state built its proposal around a $1.1 billion award and its new limit is $700 million, Florida will have to rethink its plan, he said.

“It’s a pretty significant reduction,” Mr. Smith said. Delaware’s $100 million prize gave that state $807 for each of its 124,000 students, Mr. Smith said. With the new cap, Florida can aspire to receive only $266 for each student, he said.