The latest weekly sampling by the Office for National Statistics suggest 473,800 people in Scotland had the virus in the week ending 20 March.
It came in the same week Covid patients in hospitals reached a record high.
And NHS Lanarkshire warned its three acute hospitals – Hairmyres, Wishaw and Monklands – were beyond full.
Medics said they were regularly running with capacity over 100% and they urged people to attend its accident and emergency departments only if their condition was urgent or life-threatening.
The surge in case numbers has been attributed to a more infectious Omicron sub-variant, BA.2.
The one in 11 rate was the UK’s highest, with England and Wales reporting one in 16 and one in 17 in Northern Ireland.
The number of cases in Scotland has risen from 376,300 last week – when one in 14 had the virus – to 473,800 people in this week’s data.
Meanwhile, there were 2,326 hospital patients with the virus on Friday – the fifth day in a row that Covid patient numbers hit a record high.