According to the new data, around one in 3,900 people were infected with Covid-19 at any point between June 22 and July 5. That equated to an estimated average of 14,000 positive tests during that time.
The ONS has warned, however, not to read too much into the figures - stressing caution in drawing conclusions based on the low number of positive cases.
During the 14-day period of the surveillance study, there were an estimated two new coronavirus infections for every 10,000 people per week - equating to an estimated 1,700 new cases per day.
The ONS estimates are based on swab tests collected from 25,662 people - of which eight individuals from eight different households tested positive.
Modelling of the incidence rate trend over time suggests incidences of new infections appear to have decreased since mid-May and have now levelled off.
Researchers suggested a "clear downward trend since the study began on April 26".
The study also found "no evidence" of regional differences in the proportion of people testing positive for coronavirus.
Due to the low number of people testing positive, the report said "there is not enough evidence to say with confidence that there is a difference in infection rates between regions".
Coronavirus: Everything you need to know