Four others, including three police officers and a civilian woman, were wounded during the gunfire in College Station, a few blocks from Texas A&M University, CNN reported.
College Station Assistant Police Chief Scott McCollum identified the slain constable as Brian Bachmann, 41, who he said was serving an eviction notice.
McCollum said other officers who responded to the initial report of gunfire "ended up shooting the gunman."
The identity of the 65-year-old male civilian who died was not released, nor was that of the alleged gunman, though the Houston Chronicle reported it was believed to be Thomas Caffall, 35, whose name was on a forcible entry detainer filed last week for the address where the shootings occurred.
The wounded included one law enforcement officer shot in the calf, two other officers with "non-life threatening injuries" and a 55-year-old civilian woman who was in surgery, McCollum said.
Bachmann had been with the Brazos County sheriff's office since 1993, CNN said.
"He was a pillar in this community, and it's sad and tragic that we've lost him today," McCollum said of Bachmann.
The Texas A&M administration issued an emergency notice to its campus community at 12:29 p.m. alerting people there was a report of an "active shooter" and by 12:44 p.m. an updated was posted on the school's Web site that a suspected was "in custody."
The newspaper said Caffall's sister, Courtney Clark, said she could not confirm her brother was the gunman, leaving release of his identity to police.
"Our hearts go out to the families in involved," Clark said. "We are just saddened by this."
Caffall's mother, Linda Weaver, told The Huffington Post her son had mental health issues in recent years.
"The minute I saw the TV I knew it was him," she told the publication. "I've been that worried about him."