It estimates that one in 160 people in England had the virus in the week to 8 October - an increase on one in 240 the previous week.
For the same week, the ONS estimates one in 390 people had the virus in Wales and one in 250 in Northern Ireland - both an increase on the week before.
An app which tracks the Covid symptoms of four million users estimates there are more than 27,000 new cases per day in the UK.
Its latest figures, for the two weeks to 11 October, found the fastest acceleration of cases in the north west, while Scotland, Wales, London and the Midlands were also increasing, but more slowly.
Prof Tim Spector from King's College London, who founded the Covid Symptom study app, said there was no longer the "exponential increases" of a couple of weeks ago - but the data still shows "new cases continuing to rise".
The R number is the average number of people infected by each person testing positive for the virus. An R number above 1 means the epidemic is growing.
At the peak of the epidemic back in April, the R number is thought to have been around 3.
From May up until mid-August after the national lockdown, it stayed below 1, but has been rising steadily since. All the restrictions in place around the UK are now an attempt to reduce transmission of the virus between people and get it back to below 1.