Williamson suffered a left pectoral muscle injury while fielding in the second test in Wellington, which New Zealand won by 12 runs to take an unassailable 2-0 lead into the final match which starts on Saturday.
"Kane is obviously a really important player for us and with the nature of the injury we couldn't be sure he wouldn't aggravate it," said coach Gary Stead in a news release.
"He's improving every day and will continue his rehabilitation as planned. Will has been waiting in the wings for a while now and he'll get a great opportunity batting at number three in a settled line-up."
There was better news about wicketkeeper BJ Watling, who passed a fitness test on a hamstring strain on Friday and will play at Hagley Oval.
Pace bowler Tim Southee will captain the side for the first time in Williamson's absence.
(Reporting by Nick Mulvenney in Sydney, editing by Peter Rutherford)
