These groups cover the majority of those most vulnerable to Covid-19, including those over 50, unpaid carers, and those aged 16-64 with underlying health conditions.
This milestone will be reached earlier than planned following the shortening of the gap between first and second doses from 12 to eight weeks.
It is one of a series of new target dates set out by the First Minister in an update to MSPs on Tuesday.
Ms Sturgeon has already announced all adults will be given a first dose by July 18.
The next milestone will be July 26, when all over 40s are due to have received both doses.
By August 30 this will include all over 30s.
Finally, the Scottish Government aims to give all adults a second dose by September 12.
Scotland is currently giving between 17,000 and 20,000 first doses a day, and around 15,000 to 16,000 second doses.
Some 82 per cent of adults have had a first dose, while 59 per cent have had a second.
Almost 90,000 Scots in their 50s have received a first dose of vaccine but not a second, along with 50,000 people with underlying health conditions. However, some people in their 50s with underlying conditions will be counted in both groups.
Ms Sturgeon said the target date of June 27 for all over 50s to have a second dose may not take in every member of the cohort, as some people may have rescheduled their second dose appointment, and others may not turn up – though she urged them to do so.
It comes after figures published by Public Health Scotland last week revealed 500,000 vaccine appointments had been missed since February.
The Scottish Government is planning for the next phase of vaccination, Ms Sturgeon told MSPs, including possible booster jags and vaccines for under 18s.
The vaccine milestones underpin a series of announcements made about the easing of restrictions, Ms Sturgeon said.
She said: “Vaccination offers us the route out of this pandemic. We know that getting both doses provides good protection, including against the Delta variant.
"So these milestones are important and they inform the decisions I will confirm today.”