Hotham, who suffered a high ankle sprain during the France series, lasted only nine minutes on his return to the playing field in the record 43-10 defeat to South Africa last Saturday before coming off following a clash with lock Lood de Jager.
The 22-year-old posted an X-ray scan on social media showing a clean break in his fibula just above his left ankle.
New Zealand Rugby said Hotham would not feature in the home test against Australia at Eden Park on September 27 or the final match against the Wallabies in Perth a week later, both of which double as the Bledisloe Cup, the annual series contested between the trans-Tasman neighbours.
A timeline on his recovery was being assessed, NZR said.
Hotham is the third scrumhalf in Scott Robertson's squad to succumb to injury, following a foot stress fracture for Cam Roigard and a broken rib for Cortez Ratima.
Robertson said on Saturday he was hopeful Roigard and Ratima would be available for Eden Park.
In Hotham's absence in Wellington, Finlay Christie took over at scrumhalf as the All Blacks crashed to their heaviest defeat in over 120 years of tests.
New Zealand are third on the Rugby Championship table on 10 points, one behind leading Australia and level with second-placed South Africa.