The 31-year-old limped off the field during his side's shock 14-12 defeat to Japan earlier today.
After being tested by the medical team, it has now been confirmed that New Zealand will have to continue their campaign without the two-time World Cup winner.
A tweet from the Olympic team read: "NZ Sevens injury update. @SonnyBWilliams ruled out with partial Achilles rupture. Replaced by Sione Molia."
New Zealand's next match is against Great Britain tomorrow.