The move, pending the approval of the Nevada Board of Regents, came a little more than a week after Lon Kruger left UNLV following seven seasons to become Oklahoma's new coach.
Terms of the deal were not disclosed. UNLV athletic director Jim Livengood, in a statement, said no further information would be available until a news conference scheduled for 4 p.m. Monday.
The 42-year-old Rice beat out three other finalists: Minnesota Timberwolves assistant and former UNLV star Reggie Theus, St. John's assistant Mike Dunlap and former Oregon coach Ernie Kent.
Rice was interviewed Wednesday by Livengood, UNLV president Neal Smatresk and several regents, the Las Vegas Sun and Las Vegas Review-Journal reported.
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He played for coach Jerry Tarkanian as a reserve on UNLV's 1990 national championship team and 1991 Final Four team, then served as a Rebels assistant for 11 seasons under Tarkanian and three other coaches before Kruger arrived in 2004.
Tarkanian's preference was for Theus, but he did not slight Rice.
"He's a really sharp guy and a good person," Tarkanian told the Review-Journal.
Rice has 19 years of coaching experience, including 17 seasons at BYU, UNLV and Utah State. He joined BYU as an assistant in 2005 and was promoted to associate head coach in 2008. He helped lead the Cougars to the final round of 16 in the recent NCAA tournament.
"Dave has been a big part of the success of our program over the past three years," BYU coach Dave Rose said. "He is an outstanding teacher of the game and has always been part of successful programs both as a player and a coach."
In his last eight seasons at UNLV, the Rebels received seven postseason invitations and averaged 20 wins per year.
He takes over a Mountain West Conference team that went 24-9 last season and advanced to the NCAA tournament for the fourth time in five years. The team returns four starters.