They have been living in a shelter 700m (2,300ft) underground. However, the Plan B drill - the second of three which have been working simultaneously - has penetrated 624m to a workshop which can be reached by the miners.
Officials say everything needed for the rescue is now in place.
However, they still need to determine whether the miners can be winched up through the exposed rock, or if they will have to wait for the shaft to be encased with steel piping.
Huge cranes have been brought in to lower the metal casing if it is needed. Tests are expected to take hours, possibly days, and Chile's Mining Minister Laurence Golborne has warned that it will be three to eight days before the rescue mission can begin.
The layers of rock nearest to the surface are crumbly and loose and will need definitely need casing, the BBC's Tim Willcox reports from the mine.
But if the authorities decide not to encase the rescue shaft all the way down to miners, the effort to pull them out could begin within a few days.
They will drop a camera down the escape shaft to inspect the quality of the rock.
Once the tunnel is secured, the rescue team will set up a winch at the top of it and lower a specially designed escape capsule down to the miners, and only then will they be brought up to the surface.
The Chilean Health Minister Jaime Manalich says his team is ready to start the rescue as early as Tuesday.
On hearing the news, families ran up the hillside by Camp Hope waving their Chilean flags, our correspondent reports.
Relatives, many of whom have been camping near the mine since the collapse, milled around in excitement at the news. Sirens wailed in celebration.
Alicia Campos, the mother of trapped miner Daniel Herrera, said she was "very happy".
"I have a huge sense of relief because I feel I'll see my son soon," she told the BBC.
It has been a long wait for the families, our correspondent Tim Willcox reports.
For weeks they have been anxiously awaiting any snippet of news, and grabbing short opportunities to speak to their loved ones by phone.
Many families waited up all night in expectation of the breakthrough, our correspondent says.
When the rescue operation begins, a medic will be sent down the shaft initially, in a special capsule, to assess the miners. Then it is expected to take an hour to winch each man to safety.
The men are expected to be split into three groups. Some who are fit and have the most technical know-how will be chosen to go first - in case something goes wrong.
Then the weakest are expected to be brought to the surface.
A final group, including some of the strongest miners, will wait till last.