Two others were wounded in a separate shooting on Wednesday, prompting authorities to close the rank temporarily.
Police and members of the South African National Defence Force have been deployed to the site.
Despite the closure of the Palms taxi rank, minibus taxis continue to operate outside the rank, while smaller ranks in the area remain open.
The shootings come amid a wider wave of taxi-related violence across Cape Town, with incidents reported in Gugulethu, Khayelitsha, Epping and Bonteheuwel in recent weeks.
The violence is linked to disputes over lucrative routes between rival taxi associations CATA and CODETA.
Provincial government officials are expected to meet with taxi industry representatives on Thursday to discuss the escalating conflict.
Mobility MEC Isaac Sileku has warned that affected taxi ranks and routes could be shut down if the bloodshed continues, as he did last year.
Police and City of Cape Town officials are also expected to brief the media on Thursday on the violence and broader challenges facing the taxi industry.
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