The person, who lives on the border of Oldbury and Tipton, is the first to test positive for the variant in the region since two cases were found in Walsall and one in Stafford in February.
Thousands of people were tested in the nearby areas after those cases were identified and a mobile testing unit is now being set up at the Pure Gym car park in Dudley Port.
People aged 16 and over who live in specific streets in the Oldbury and Tipton border area are being urged to take a rapid Covid-19 swab test, even if they are not showing symptoms.
It includes specific streets within the wider DY4 7, DY4 8 and B69 2 postcode areas, in an area loosely between Birmingham Canal in the north, New Birmingham Road in the south, Coneygree Road in the west and Vaughan Trading Estate in the east.
The mobile testing unit at the Pure Gym car park will be open between 10am and 3pm on Wednesday, for walk-in tests and booked drive-in tests. A booking system for drive-in appointments will be available on Sandwell Council's website.
From Thursday, the mobile testing unit will be open between 9am and 3pm seven days a week for walk-in tests and booked drive-in appointments.
Sandwell Council has confirmed the South African variant case is not linked to international travel.
Councillor Maria Crompton, the council's deputy leader, said: "I know this news will cause concern. Our priority is to keep our communities and especially our most vulnerable residents safe.
“We are making additional, targeted testing easily available in these specific areas, so we can monitor and control this new variant.
“We are strongly encouraging people to get tested even if they don’t have symptoms.”
Dr Lisa McNally, Sandwell's director of public health, said: "The more cases of the variant we can find, the better chance we have of suppressing it.
“If you are asked to be tested please, please do take up the offer, it’s quick easy and painless and you’ll be saving lives by doing so.
“Please help to keep your community safe by getting tested so we can protect you and your loved ones.”
There are no additional restrictions or lockdown measures for those living in the Oldbury and Tipton border area – and people should continue to follow the national guidelines.
Anyone who tests positive must self-isolate immediately and pass on details of their contacts to NHS Test & Trace.
Katie Spence, deputy director at Public Health England Midlands, said: "The UK has one of the best genomic systems in the world which has allowed us to detect the variant first identified in South Africa here in the Oldbury/Tipton border area.
“I urge everyone offered a test to take it up to help us to monitor the virus in our communities and to help suppress and control the spread of this variant.
“The most important thing is that people continue to follow the guidance that is in place – limit your number of contacts, wash your hands regularly and thoroughly, keep your distance and cover your face. If you test positive by any method, you must isolate to stop the spread of the virus.”
Dr Susan Hopkins, chief medical advisor for NHS Test and Trace, added: “As part of our proactive sequencing work, we have identified cases of Covid-19 variants in a number of areas across England.
“A small proportion of these cases have no link to international travel suggesting that there are some cases in the community.
“PHE and NHS Test and Trace are monitoring the situation closely and appropriate public health measures such as increased testing and contact tracing are being put in place where necessary.”
West Bromwich West MP Shaun Bailey said: "The discovery of the South African variant on the border of Oldbury and Tipton is, of course, concerning for residents and the wider community.
"I would therefore strongly encourage everyone within specific streets in the wider DY4 7, DY4 8 and B69 2 postcode areas, to get a test, whether you are experiencing symptoms or not."
To find out if you live in the area of targeted testing, visit sandwell.gov.uk/testsandwell
For anyone who is not online or who has questions about the targeted testing in the Oldbury/Tipton border area, call the Healthy Sandwell team on 0800 011 4656 for advice.