Peers have backed down in their fight against ministers over "fast-track" extraditions to the US.
The NatWest Three were extradited to the US earlier this year
They had argued the current system was unfair, saying it was easier for the US to extradite Britons than vice-versa.
But opposition peers have ended their effort to remove the US from countries that do not need "prima facie" evidence to extradite from the UK.
Home Secretary John Reid called the decision a "strong message in support of justice".
The end of the battle means the current parliamentary session will probably end on Wednesday morning.
The Liberal Democrats fought to re-instate a clause into the Police and Justice Bill ensuring that a judge could not order extradition to the US unless it was deemed "in the interest of justice" for the trial to take place abroad.
But without Conservative support, the proposal was defeated by 96 votes to 174.
Moves to amend the extradition terms between the US and UK followed the controversy over the extradition of three UK bankers - the "NatWest Three" - on fraud charges.
Peers have twice inflicted defeats over the issue and last week 14 Labour backbenchers rebelled over it.
On Monday MPs overturned the Lords amendments for the second time.
Mr Reid had earlier argued that if the treaty was blocked by the amendments it could result in "very serious offenders and suspected offenders escaping justice".
Following the Lords vote, he said: "Parliament has sent a strong message in support of justice today in allowing the UK/US extradition treaty to proceed to ratification.
"This will bring real benefits to British victims of crime and help ensure that serious criminals will not be able to escape justice by hiding abroad."