By placing Norton on the reserve/non-football injury list, the team is able to pay the 22-year-old his full salary.
According to the Florida Highway Patrol, Norton's Ford F-250 crashed into a concrete barrier on State Road 836 in the early hours of July 4.
The University of Miami product underwent at least six surgeries before he was released from the hospital last week.
"I just want to give a big thank-you, first of all, to God for me still being here. Second of all, to Jackson Medical Center. They did a great job. They took great care of me while I was in here," Norton told reporters outside the hospital. "Next, I would like to thank the Miami Dolphins family. They've been crazy with how much help they've been willing to give and, you know, they really didn't have to."
The hope is that Norton will be able to get a prosthetic arm as he transitions to life after football. He confirmed last week that his playing career is over after the accident.
Dolphins coach Brian Flores has been a frequent visitor. General manager Chris Grier and several of Norton's teammates have visited him, as well, giving their support.
All of Norton's medical bills will be covered by the NFL's and Dolphins' insurance policies.
Norton was cited as at fault for the July 4 accident with an improper lane change, due to his pulling in front of another vehicle and making slight contact with it before crashing into a concrete barrier, according to a copy of the Florida Highway Patrol accident report obtained by ESPN. According to the report, alcohol and drugs were not suspected as a factor in the accident for either driver.
Norton was selected in the seventh round of the 2018 NFL draft by the Carolina Panthers, and the Dolphins signed him off the Panthers' practice squad in December. He was expected to compete for a spot on the 53-man roster this season.
ESPN's Cameron Wolfe contributed to this report.