The results of the pilot study suggest an average of one in 370 people, or 0.27% of the population - excluding health and care workers - were infected with the virus over the period.
However, among people working in patient-facing healthcare and resident-facing social care roles, 1.33% tested positive for Covid-19 during the time frame.
This includes NHS professionals, such as nurses and doctors, as well as social care workers, such as nursing home or home care workers.
For those reporting not working in these roles, 0.22% tested positive.
The survey is designed to help track the current extent of infection and transmission of Covid-19 among the community population - specifically private households.
As such, the survey excludes infections reported in hospitals, care homes or other institutional settings.
The ONS said there was "no evidence" of differences in the proportions of people testing positive for coronavirus between different age categories.
The estimated percentages of the sample who tested positive for Covid-19, broken down by age, were: 2-19 years old: 0.32%; 20-40 years old: 0.26%; 50 to 69 years old: 0.32%; 70 years and over: 0.23%.
The number of swab tests collected during the survey
The figures are based on 10,705 swab tests collected from people participating in the recently launched coronavirus (Covid-19) infection survey, which is being produced by the ONS in partnership with the University of Oxford, the University of Manchester, Public Health England and the Wellcome Trust.
Out of the 10,705 tests, 33 individuals in 30 households tested positive for the virus.
The ONS is also running a bigger long-term study to track the spread of Covid-19 in the general population.
The wider study, which will include up to 300,000 people, includes antibody testing to help understand how many people have had Covid-19 in the past.
Coronavirus: Everything you need to know