Rice, who had been suffering from a brain tumour, played most of his cricket during South Africa's 20-year isolation from international cricket. Rice succumbed following a long battle five days after his 66th birthday.
"Clive was our first captain and we knew him to be a great fighter all his life," said CSA Chief Executive Haroon Lorgat.
"Even during his last few years he put up a typically courageous and inspirational fight against the illness that had threatened him for a lengthy period of time.
Rice's best playing days coincided almost identically with South Africa's isolation from international cricket because of apartheid, which divided the sport on racial lines in the country.
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He was selected for the cancelled tour to Australia in 1971-72 at the age of 22 and was 42 when he led his country to India on the 'Friendship Tour' as democracy dawned with the release of Nelson Mandela from prison.
As their captain he led them to the county championship in 1981 for the first time since 1929 and formed an internationally respected new ball partnership with Sir Richard Hadlee of New Zealand.
During his career he made more than 39 000 runs in first-class and limited overs cricket and took more than 1 500 wickets.
As a mark of respect the Proteas will wear black armbands in the Test match against Bangladesh starting Thursday.
Rice was full of praise for the Indian medical specialists.