"Our number one priority is the health and safety of the England team and management group, and the correct decision was made following discussions between the two boards and respective medical teams.
"Things are inevitably a bit more relaxed here as there is no hotel on site, so both sets of players are staying in a hotel just five minutes away.
South African backroom staff pack up after the postponement of the first one-day international
The game at Newlands was due to begin at 11am on Friday but was called off just an hour before play was set to begin. It will now take place on Sunday.
"The England party will remain at its base in Cape Town on Friday and Saturday and we are hopeful that the three-match series will be played before we depart next Thursday starting with the match at Paarl on Sunday."
In a joint statement with the England and Wales Cricket Board (ECB), CSA added: "In the interests of the safety and well-being of both teams, match officials and all involved in the match, the acting CEO of CSA, Kugandrie Govender, as well as the CEO of the ECB, Tom Harrison, have agreed to postpone the first fixture to Sunday."
A statement from Cricket South Africa (CSA) confirmed one member of their squad returned a positive Covid-19 test following their latest round of testing on Thursday.
"There is movement between venue and accommodation, which the ECB got lucky with in a sense in that the Ageas Bowl and Emirates Old Trafford had hotels on site and everything could be clamped down very efficiently."
Last Updated: 04/12/20 4:28pm
Sky Sports Cricket pundit Mike Atherton said that the positive test in Cape Town demonstrated the effectiveness of the bio-secure bubbles set up in England earlier this summer, when no player fell ill throughout series between the hosts and West Indies, Ireland, Australia and Pakistan.
Watch the first match of the series between South Africa and England live on Sky Sports Cricket from 7.30am on Sunday.
Ashley Giles, managing director of England men's cricket, issued a brief statement following the postponement declaring himself "hopeful" that the matches could be successfully staged.
"Fingers crossed, first of all, that the player is ok and then they'll try to work out the proximities - who he's been with and presumably they will test everybody else too. Let's hope we can get the three games on," the former England captain said.
Fellow pundit Nasser Hussain added: "Sometimes these things just happen. They've taken absolutely the right procedure - you can't risk anyone else. South Africa had already had two players who had gone down with Covid-19.
"We wish whoever it is gets well as soon as possible. It's just part of the world that we live in at the moment. In fact, in South Africa, there has been - like in most parts of the world - a spike. They're talking about closing things down."
"On behalf of the England touring party and the ECB, I would like to thank Cricket South Africa for their collaborative approach to postponing today's one-day international," he said.
"What it does show is the remarkable job that the ECB did in the English summer at the height of the pandemic, to get all of those games on with no players testing positive for Covid-19.