The 1,201 cases reported Thursday is the seventh-straight day with cases topping 1,000.
In the last week, Dallas County has added 7,771 new cases of the virus.
The county has now accumulated more than 30,300 cases of the virus since testing began in March.
The increase in cases comes as the state's positivity rate, the percentage of people testing positive for the virus, has been sustained well over 10% for more than two weeks and grew to just above 15% on Wednesday.