Chilean President Sebastian Pinera has told the 33 men trapped in a mine in the north of the country every effort is being made to free them.
In a telephone conversation with the shift leader, Luis Urzua, Mr Pinera said the whole of Chile was with them.
The miners urged all efforts to be made to free them by 18 September, Chile's Independence Day.
The men, trapped since 5 August, have not been told that their rescue could take up to four months.
"Mr President, we need you to be strong and to rescue us as soon as possible. Don't abandon us," Mr Urzua said.
Mr Pinera, indicating that their rescue would take "some time", said: "You will not be left alone, you have not been alone. The government is with you all, the entire country is with you all."
He added: "We will knock on every door, we will search for every bit of technology, equipment, every expert, so we can get you out alive.
"Because for us, being able to rescue you alive is a promise that signifies the soul, the love and the spirit of the Chilean people."
Mr Pinera also promised that the men's families - many of whom are camped out at the surface - would be looked after.
The Chilean government has approached Nasa for advice on helping the men deal with being confined and isolated for a long time.
The BBC's Gideon Long, at the mine, says officials are considering sending playing cards and games to the miners to give them something to occupy their time.
They are also investigating ways of simulating day and night, to give the miners a sense of normality, says our correspondent.
Specialist equipment has arrived at the San Jose mine in Copiapo, which is some 725km (450 miles) north of Santiago.
It will bore a shaft down to the men, who are located some 700m (2,300ft) underground.
Andres Sougarret, the engineer in charge of the operation, is planning to drill a new tunnel, then widen it so the men can be pulled one by one to the surface.
He has estimated that it could take three to four months.
However, US mining expert Davitt McAteer said that was a "conservative" estimate.
"We ought to be able to get them out in a period of weeks, not months," he told the Associated Press.
The miners, who spent 17 days surviving on emergency supplies designed for a couple of days, have been receiving glucose and rehydration tablets.
Other supplies included small lights, eye patches and medicine.
Chilean officials say two of the miners have some medical experience and would help people above to monitor the health of their colleagues.
The miners were trapped when the main access tunnel collapsed. They are located 7km (4.5 miles) into the winding mine, where they are sheltering in a 50 sq m side-chamber off one of the main passages.
Days of growing despair turned to jubilation on Sunday when rescuers made contact with the miners via a probe lowered into the mine.
One of the men, Mario Gomez, who wrote a note indicating that they were alive, has become a symbol of their resilience.
His wife said that her husband had sent her an emotional letter with the note.
"It said that he loves me. I've never received words from him like that, even when we were engaged, he has never been a romantic," Liliana Ramirez told the AFP news agency.
The miners' families are maintaining a vigil outside the mine. They have been sending personal messages to the men but have avoided mentioning the time it may take to rescue them.