There was no immediate word from police about a possible motive.
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The attack happened just after 1:30 p.m. The suspect has been identified as Kendrex J. White. He is also a UT student.
UT President Greg Fenves talks to the media about the fatal stabbing on UT campus
Student Rachel Prichett said she was standing in line at a food truck outside a gym when she saw a man with a large knife approach the person standing behind her.
"The guy was standing next to me," Prichett said. "He grabbed him by the shoulder and shoved the knife in it. I just started running as fast as I could."
University of Texas Police Chief David Carter briefs the media on the fatal stabbing on campus
Another student, Ray Arredondo, said he was walking to his car when a mass of students near the gym started running.
"They were just screaming, 'Run! Get out of here!'" Arredondo said.
Travis County Emergency Medical Services tweeted that one person died at the scene near the gym. The others were taken to the hospital with potentially serious wounds. There were reports of additional patients with non-life-threatening injuries, the agency said.
Lindsey Clark said she saw the suspect get tackled by police as he was running toward the entrance of Jester Hall, a complex of dormitories and classrooms. She described him as wearing a bandanna and gray sweatshirt and said he appeared quiet and subdued as police held him on the ground.
"You could see and hear people running and screaming: 'There he is!' before he was tackled by officers," Clark said.
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University Police Chief David Carter said it would be "premature" to discuss the suspect's motive and "what was going through his mind."
Arredondo later saw what looked like CPR being performed on someone outside the front door of the gym. Another student was sitting on a bench being treated for cuts to the head or neck, he said.
Authorities cordoned off the scene as a large contingent of state and local police, including officers in helicopters, swarmed the area.
All classes and events have been cancelled for the rest of the day.
The university sent an alert to everyone on campus saying that there was criminal activity with injury.
Governor Greg Abbott offered his prayers to everyone in Austin.
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"Praying for all those affected by this heinous attack and for the UT-Austin community," Abbott tweeted.
The University of Texas is blocks from downtown Austin and the Texas Capitol and is one of the nation's largest universities.
The attack occurred in the central campus, just a short walk from the administration building and the landmark clock tower that was the scene of a mass shooting in 1966.
The attack came only a few days after a 19-year-old man armed with a machete wounded two people at a university coffee shop in Lexington, Kentucky.
In the April 28 attack, the assailant at Transylvania University asked about the political affiliations of people at the shop. He was arrested and charged with assault. The victims' wounds were not life-threatening, authorities said.
This story is developing. Stay with Eyewitness News for updates on air and online.
The Associated Press contributed to this report.
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