A total of 264 deaths registered in the seven days to June 17 mentioned coronavirus on the death certificate, according to the Office for National Statistics (ONS).
This is down slightly from 284 deaths in the previous week and remains well below the 1,125 recorded in the peak week of the Omicron BA.2 wave of infections earlier in the year.
The current wave is being driven by the newer variants BA.4 and BA.5, which now make up more than half of all new Covid-19 cases in England.
The BA.5 variant is thought to be growing approximately 35% faster than BA.2, while BA.4 is growing 19% faster – meaning it is likely that BA.5 will soon become the dominant Covid-19 variant in the country.
But there is “currently no evidence” that the two variants cause more serious illness than previous variants, according to early research from the UK Health Security Agency (UKHSA).
Infections in all four nations of the UK are rising, with levels in England back to where they were at the end of April.
The number of people in hospital in England who have tested positive for Covid-19 is also continuing to increase, in further evidence of the growing prevalence of the virus.
Some 7,822 patients in England had Covid-19 on June 27, up 37% on the previous week, NHS figures show.
It is the highest total for nearly two months but is still some way below the peak of 16,600 patients during the Omicron BA.2 wave.
Most hospital patients who test positive for Covid-19 are being treated primarily for something else, rather than the virus.
But they will need to be kept isolated from those patients who do not have Covid, putting extra pressure on hospital staff.
UKHSA chief medical adviser Professor Susan Hopkins said: “It is clear that the increasing prevalence of Omicron BA.4 and BA.5 are significantly increasing the case numbers we have observed in recent weeks.
“We have seen a rise in hospital admissions in line with community infections but vaccinations are continuing to keep ICU admissions and deaths at low levels.”
Death registrations are not likely to show any clear impact from the current rise in infections for several weeks, given the time it takes someone with the virus to become seriously ill.
In Wales, 417 hospital patients with Covid-19 were recorded on June 24, up 41% from the previous week.
Infections in Wales have climbed to levels last seen in early May.