It wasn't immediately clear where the rocket landed, or what condition it was in.
The 20-foot SpaceLoft XL rocket, set up by Connecticut-based UP Aerospace, was loaded with 50 experimental and other payloads for a 13-minute suborbital flight about 70 miles above Earth.
The flight took off at 4:14 p.m. EDT after hours of delay. The launch was originally set for 9:30 a.m. EDT, but was pushed back because a transponder was not communicating with the missile range.
The payload included school experiments from New Mexico and other states.
Among the experiments was one from Farnsworth Aerospace Magnet School in St. Paul, Minn., which sent two students to watch the launch. Their experiment included two digital and two analog watches to analyze how the pressure of space launch affects timepieces.
Eric Knight, CEO of Farmington, Conn.-based UP Aerospace, said last week that the flight marks the chance for the public to have "direct access to space." Payload space on one of his rockets ranges from a few hundred dollars to tens of thousands of dollars, depending on size, he said.
Several other UP Aerospace flights are set later this year, including an Oct. 21 flight expected to carry the ashes of James Doohan, who gained fame as chief engineer Montgomery "Scotty" Scott on the original "Star Trek" TV series, and Mercury astronaut Gordon Cooper.
The Upham site also is the planned home of a state-built $225 million spaceport. UP Aerospace's rocket was launching from a temporary pad.
Richard Branson, founder of the Virgin Group, announced plans last year to base his space tourism company, Virgin Galactic, in New Mexico and to launch flights from the spaceport by the end of the decade.