Some 666 deaths registered in the week ending October 8 mentioned Covid-19 on the death certificate, according to the Office for National Statistics (ONS).
This is down 15% – the biggest week-on-week fall since the week ending May 21, when the figure fell by 29%.
It is also the lowest number of deaths since the week to September 3.
Around one in 16 (6.2%) of all deaths registered in the latest week mentioned Covid-19 on the death certificate.
The number of deaths involving Covid-19 decreased in all nine English regions, but rose in Wales from 64 to 81.
Since mid-June the number of deaths registered each week has followed a broadly upward trend, reflecting the impact of the third wave of Covid-19.
The latest figures suggest this trend may have come to a halt, but it is too soon to know whether this is the start of a clear downward path.
Deaths during the third wave of the virus have never reached the levels seen at the peak of the second wave.
Some 8,433 deaths involving Covid-19 were registered in England and Wales in the week to January 29.
The relatively low number of deaths in the third wave, in comparison, reflects the success of the rollout of coronavirus vaccines.
But while registered deaths involving Covid-19 are falling overall, there are still more people dying than normal for this time of the year.
Some 10,807 deaths were registered in England and Wales in the week to October 8 – 297 more deaths than the previous week and 13.1% above the five-year average.
It is the 14th week in a row where the ONS has reported extra deaths, or “excess deaths”.
In total, 164,285 deaths have occurred in the UK where Covid-19 was mentioned on the death certificate, the ONS said.
This includes 43,564 care home residents in England and Wales who have had Covid-19 recorded on their death certificate since the pandemic began.
The ONS figures cover deaths of care home residents in all settings, not just in care homes.
Prof Kevin McConway, emeritus professor of applied statistics at The Open University, said it is encouraging that weekly registered coronavirus deaths are falling.
He said “the position isn’t wonderful”, but added that “the trend is in the right direction”.
But he said it is “somewhat concerning” that registered deaths from all causes are above the five-year average.
He continued: “In England and Wales, there have been excess deaths compared to the five-year average for 14 straight weeks now, and the excess number has been consistently quite a lot larger than the numbers of registered deaths involving Covid-19.
“We still don’t have comprehensive data on what is causing these excess deaths.”