Texas Runoff Set in G.O.P. Race for Senate AUSTIN, Tex. - In a heated race for the open United States Senate seat in Texas, Lt. Gov. David Dewhurst failed to win enough votes in the Republican primary on Tuesday to avert a runoff with Ted Cruz, a Houston lawyer who cast the election as a battle between the moderate Republican establishment and Tea Party conservatives. Mr. Dewhurst, one of the longest-serving lieutenant governors in state history, was the front-runner in a field of nine primary candidates seeking to replace Senator Kay Bailey Hutchison, a Republican who is retiring after serving three terms. Mr. Dewhurst, a rancher and businessman who pumped more than $9 million of his fortune into his campaign and who was backed by Gov. Rick Perry, fell a few points short of exceeding 50 percent of the vote and faces a July 31 runoff with Mr. Cruz, who placed second, according to The Associated Press. Tom Leppert, a former mayor of Dallas, was a distant third. In a speech in Houston to a crowd of cheering supporters on Tuesday night, Mr. Dewhurst focused not on Mr. Cruz, but on the Obama administration. "Tonight we Texas Republicans said, "We're going to repeal Obamacare," " Mr. Dewhurst said. Tonight you showed that we're all ready to stop the out-of-control spending by Washington. Mr. Cruz, a former state solicitor general who was outspent by both Mr. Dewhurst and Mr. Leppert, has been a darling of the Tea Party movement, and has often been compared to a fellow Cuban-American conservative, Senator Marco Rubio of Florida. Speaking to his own supporters in Houston, Mr. Cruz called the results a victory. "This is an incredible, unbelievable night, and this is a testament to every single one of you," he said. Nobody in the state thought we could do it, and together, we did. As Texas solicitor general, Mr. Cruz won cases before the United States Supreme Court that, among other things, defended the right to have a monument to the Ten Commandments at the state Capitol. He received support from Sarah Palin, the former governor of Alaska, and the former presidential candidate Rick Santorum, endorsements that Mr. Dewhurst cast as outside meddling in state politics. In the final days before the primary, Mr. Dewhurst's campaign produced a radio advertisement that accused Mr. Cruz of helping to run "two national organizations that have been leading the push to give amnesty to illegal immigrants." The ad was criticized by political observers and by one of the two groups, the Hispanic Leadership Fund. The group's president, Mario H. Lopez, said that Mr. Cruz had never held any leadership role in the organization, and that it did not support amnesty. Mr. Cruz's campaign said he had always opposed amnesty and called the ad "a transparent attempt at race-baiting." Statewide, turnout was relatively low Tuesday, coming the day after Memorial Day and after a federal three-judge panel in San Antonio twice pushed back the primary date because of a redistricting dispute between the state and minority groups. The primary was originally set for Super Tuesday, March 6, before the judges delayed the date to April 3 and then to Tuesday. It was one of the last primaries in the country, weakening the state's voice in the Republican presidential race. Mr. Cruz, however, said the delayed primary had helped him by giving him more time to campaign against Mr. Dewhurst, who since 2003 has held what many regard as the most powerful position in state government.