No play took place after rain fell during the lunch break on the second day on Saturday, when New Zealand were on 15 for two in reply to South Africa's first innings total of 263.
Rain fell throughout the rest of Saturday and into early Sunday morning and further play was ruled out on days three, four and five because of a wet, soft outfield which the umpires decided would be dangerous for players.
The series now hinges on the second and final Test starting at Centurion on Saturday.
Cricket South Africa chief executive Haroon Lorgat in part blamed unseasonal rain and a re-laid outfield for the loss of play at the Kingsmead ground.
He said on Monday the outfield had been relaid in June following complaints by players about conditions during a Twenty20 international and a one-day international there in August 2015, also involving New Zealand.
"However, the unseasonal rain at this time of year impacted our preparations of the newly laid outfield," he added.
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