"There are dozens of other people that helped me and I appreciate them all," she wrote. "Thank you."
She suffered multiple serious injuries, but none of them appeared to be life-threatening. Medics airlifted her to an Albuquerque area hospital for treatment.
"I am alive," Williams wrote. "Unfortunately, the bear is not."
Officials with the New Mexico Department of Game and Fish expressed regret Sunday over having to kill the bear, which was tracked by conservation officers and found near the attack site. However, New Mexico law requires that "any wild animal that attacks or bites a human be euthanized and tested for rabies."
The bear had been part of a study involving wild bears, according to the Department of Game and Fish, and was wearing a GPS collar. The collar helped authorities confirm that the bear that was euthanized was the one responsible for Williams' injuries.
"We are thankful that the injuries sustained by the victim were not worse and are hopeful that she is able to recover quickly," said Alexandra Sandoval, director of the state's Department of Game and Fish.
Conservation officers said they were unable to capture the bear's three cubs on Sunday. Officers said they would continue efforts to catch the young cubs, who they plan to send to the New Mexico Wildlife Center in Espanola for care.