Fifth seed Hampton, who had made it into the last four in Auckland for the second year in a row, picked up a hip injury during training on Friday morning.
Williams, who is unseeded, will meet either Serbia's Ana Ivanovic or Belgian Kirsten Flipkens, the No.2 and No.3 seeds, in the title decider on Saturday.
After making the semi-finals, Hampton had talked about wanting to face Fed Cup teammate Williams on court for the first time, saying she had nothing but respect for the former world No.1 and seven-time grand slam winner.
She expressed her disappointment that the chance would now not come this week.
"It's incredibly disappointing," she said.
"I would have loved the opportunity to have played such a great champion."