Mrs Liebenberg's application for residence has been declined because she "did not meet the health requirements". Her request for a medical waiver was unsuccessful.
Mrs Liebenberg has several medical conditions, including restrictive lung disease, moderate aortic stenosis and hypertension.
"Her valvular condition seems to be the most pressing issue ... it is well known that cardiac services are overburdened," said immigration manager Ruth Meek in the rejection letter.
"Her heart condition is deemed to impose significant costs to New Zealand's health system. I consider that the future burden on New Zealand's health services outweighs the applicant's circumstances."
Immigration New Zealand has told Mrs Liebenberg she has to leave the country by August 31, despite her doctor having written to the agency saying she could not travel.
"With her poor functional status and requirement for oxygen, she is not fit to fly," said cardiologist Selwyn Wong.
"Her prognosis is limited in view of her severity of her lung disease, and I would support the humane view that she be allowed to stay with her daughters as her life expectancy is measured in months rather than years."
Dr Wong said that because of the severity of her illness, she "will not need expensive aortic valve intervention".
Mrs Liebenberg is staying with her younger daughter, Petro, 40, in Pukekohe, where she requires oxygen to help her breathe.
"When I made the decision to move here, I had pictured myself exploring New Zealand's vast countryside, beautiful mountains and glaciers till a ripe old age, not to find out that I will be facing life's end at 61," she said.
Mrs Liebenberg said she was very close to her two daughters as she had raised them as a single mother because her husband died when she was 21.
She said she would qualify for health funding in South Africa, but had no relatives there and her health prevented her from travelling.
"You cannot imagine what it's like to be dying in a foreign land where you are afraid of even going to the hospital because you know you owe them money," Mrs Liebenberg added.
"I did not intend to be a burden to New Zealand's health system, it's just bad luck.
"My only wish is that I could die with dignity, with family beside me, wherever that may be."