The variant, first identified in India, now accounts for almost all cases across the UK, experts said.
Across the UK, cases have risen by almost 80 per cent, with 75,953 now identified.
The majority are in England, while 254 cases are in Norther Ireland and 184 in Wales.
Dr Jenny Harries, chief executive of the UK Health Security Agency, said the increase across the UK is being driven by younger age groups.
Dr Harries said: “Cases are rising rapidly across the country and the Delta variant is now dominant.
“The increase is primarily in younger age groups, a large proportion of which were unvaccinated but are now being invited to receive the vaccine.”
Almost half of those in hospital with Covid in Scotland are under the age of 25.
Around this time last year, that age group accounted for fewer than 5 per cent of recorded cases.
Public Health Scotland figures show there were 27 people under 30 in hospital with Covid in the week to June 1, including eight children under ten.
However, these figures are not a complete measure, as they include anyone admitted to hospital for a different reason, but who tested positive in the two weeks prior to admission, or caught the virus while in hospital.
All those over 18 in Scotland will receive a vaccination appointment by the end of next week, the Scottish Government has said.
Dr Harries said: “It is encouraging to see that hospitalisations and deaths are not rising at the same rate, but we will continue to monitor it closely.
“The vaccination programme and the care that we are all taking to follow the guidance are continuing to save lives.
“Please make sure that you come forward to receive both doses of the vaccine as soon as you are eligible. Don’t drop your guard – practice hands, face, space, fresh air at all times.”