Both players completed physicals and have signed contracts.
Gonzalez, a right-hander from Oral Roberts University, and Demeritte, from Winder-Barrow (Ga.) High School, have been assigned to Spokane of the Northwest League and Arizona League, respectively.
According to sources, Gonzalez agreed to a signing bonus of $2.2 million and Demeritte to a bonus of $1.9 million, both over slot for their pick value under the current Collective Bargaining Agreement.
The Rangers also signed second-round pick Akeem Bostick, a right-handed pitcher from West Florence (S.C.) High School.
Bostick signed for $520,600, which is well under the $899,400 slotted for the 62 pick.
The 21-year-old Gonzalez, who went 9-5 with a 1.83 ERA in 113 1/3 innings, said he is ready to keep pitching after completing his junior season. He doesn't need a break.
"I'm ready to go," Gonzalez said. "I'm strong. I threw a bullpen Friday and I'm ready to go. No time off."
Demeritte, 18, who batted .404 with 12 home runs and 37 RBIs as a senior for Winder-Barrow High, was taken with the draft pick the Rangers received as compensation for losing Josh Hamilton to free agency.
Demeritte said he isn't concerned with being the player the Rangers took with the Hamilton pick.
"I just want to get in and do what I do best," Demeritte said. "I'm not worried about that. I'm just ready to make my name."
The 18-year-old Bostick went 10-0 with a 0.62 ERA as a senior for West Florence High. He is a three-sport athlete, having committed to play baseball as well as wide receiver at Georgia Southern.
Bostick, whose cousin Brandon is a tight end for the Green Bay Packers, said choosing baseball was an easy decision.
"Baseball has always been my primary sport," Bostick said.
The Rangers also signed three more picks -- Liberty University center fielder Ryan Cordell, an 11th-round pick; Howard College right-hander David Gates, an 18th-round pick; and Tennessee Wesleyan College right-hander Justin Sprenger, a 29th-round pick.
The Rangers have signed or agreed to terms with 11 of their first 13 picks. Overall, they have reached agreements with 18 of their 41 draft selections.
Todd Wills is a contributor to ESPNDallas.com.