BOTH unions have suspended the 48-hour Tube strike which was due to begin today.
The TSSA union earlier confirmed that it would no longer take industrial action over ticket office closures, saying it had reached an agreement with London Underground.
And Bob Crow form the RMT union said in a statement: "After two days of intensive and detailed discussions through the offices of ACAS we have now received proposals that halt the implementation of the job cuts set out in the HR1 document which gives us the opportunity to discuss all of the issues away from the pressure cooker.
"We now have a golden opportunity to look again in detail at all of the concerns we have raised about the impact of the cuts on our members and the services that they provide to Londoners. That is exactly what we have been calling for throughout this dispute."
He said: "It is unfortunate that we were forced and provoked into a dispute that we never wanted and we are now in a position to move on with the clear understanding that our action is suspended but if there is any further attempt to impose change from above the action will go back on."
A TSSA spokesman said: "We have now agreed a process where all our serious concerns over safety and job losses will be seriously addressed through the normal channels.
"We are obviously pleased that we have agreed this process."
John Woods, deputy chief conciliator at Acas, said: "We welcome the news that the proposed industrial action has been withdrawn. We want to thank all the parties involved for their hard work and commitment over 10 days of intensive talks with Acas."
The Federation of Small Businesses estimated that last week's action cost small firms in the capital £600 million in lost working hours, business and productivity.
LU managing director Mike Brown said: "We welcome the constructive talks at Acas today and await confirmation from the RMT and TSSA that their strike action is suspended.
"We have always said that we want the unions to engage fully with us, to help shape our proposals for the future of the Tube.
"The hard work of both the LU and union negotiating teams and the progress we have made at Acas over the last few days means we can now do that without further unnecessary disruption to Londoners."
LU repeated that it planned to have staff based in ticket halls, on gate lines and on platforms rather than in ticket offices, adding that changes would be implemented without compulsory redundancies.
The Mayor of London, Boris Johnson, said: "I'm pleased the TSSA and the RMT have agreed to call off their planned strike following talks with Transport for London. It means further unnecessary disruption to London and Londoners has been averted.
"TfL's negotiators have been ready since November to discuss the detail around ticket office closures and wider modernisation of the Tube. It's welcome news that the unions appear to recognise that, and will return to full and substantive discussions with TfL between now and the end of the consultation period in early April.
"Modernisation is essential if we are to properly serve the millions of Londoners who rely on the Tube every day, and who expect a bigger, better service that offers value for money whilst protecting future investment.
"It is essential that our hard-working and dedicated staff, who are vital to the delivery of that vision, understand the changes we are proposing.
"Sitting down to discuss those proposals, free from the prospect of strike action, was always the only sensible way forward. I'm grateful to TfL's negotiating team and pleased the unions agree this is the right way forward."
Val Shawcross, leader of the Labour group on the London Assembly, said: "We now need the Mayor to show some real leadership and launch a full public consultation on his proposed cuts to ticket offices and station staffing levels.
"There are some interesting ideas in TfL's plans and modernisation is needed, but Londoners should be given the chance to have their say and improve them."