Deputy First Minister Nicola Sturgeon said it would bring the Scottish rail network "into line with the most forward-thinking transport systems in the world".
The current travelling times for the trip are 50-55 minutes. A £650million programme to electrify the whole route is already under way and is expected to reduce journey times by 10 minutes.
But the plan unveiled today is separate because it will require building a completely new line.
The projected costs or exact details of the route have not yet been revealed.
Ms Sturgeon made the announcement as politicians and railway experts gathered in Glasgow for a two-day conference on the benefits of High Speed Rail and the need for Scotland to be included.
High Speed Rail is the UK Government's plan for a fast route from London via Birmingham, which only exist for as far north as Manchester and Leeds.
Following today's announcement, the Scottish Government, with council leaders in Glasgow and Edinburgh, will lobby UK ministers to include Scotland in the High Speed Rail plans.
It is planned for the new Glasgow-Edinburgh route to be open before the London-Birmingham link is complete in 2026.
If the Westminster UK High Speed2 plan is to eventually come to Scotland it will be at 2034 at the earliest.
But Ms Sturgeon said Scotland was not able to wait for Westminster to act.
She said: "The Scottish Government is happy and proud to take the first steps to ensure Scotland has a high speed connection bringing us into line with the most forward-thinking transport systems in the world.
"We now know that, within just 12 years, we could build a line that will see journey times between our two major cities cut to less than half an hour."
Gordon Matheson, leader of Glasgow City Council, welcomed Ms Sturgeon's announcement.
He said: "I have long argued that bringing High Speed Rail to Glasgow and Edinburgh is vital for Scotland's economy.
"Glasgow City Council is committed to working with the Scottish Government to bring high speed rail to Scotland at the earliest opportunity."
The Fast Track Scotland report last year found a high speed line from London would benefit Scotland by £24.8billion.