Home Secretary James Cleverly (R) signed the new treaty with Rwanda's foreign affairs minister (L)
In the first vote of its kind, the House of Lords has called for the new UK-Rwanda treaty to be delayed until Kigali improves its asylum procedures.
Peers backed a non-binding motion urging the government not to ratify the treaty by 214 votes to 171.
The treaty, signed in December, is central to Rishi Sunak's scheme to deport asylum seekers to Rwanda but is separate to the Rwanda Safety Bill.
Foreign Secretary Lord Cameron urged his fellow peers to back the bill.
"It's not acceptable to have people travelling from a perfectly safe country - France - to another safe country - Britain - and to be able to stay, and that's what the Rwanda plan is all about," he said.
In a news conference last week, Prime Minister Rishi Sunak urged the House of Lords not to block the "will of the people" by voting down the bill.
Next week peers will begin debating the proposed legislation which aims to prevent legal challenges to deporting asylum seekers to Rwanda.
While non-binding, the vote on the treaty gives an indication of the level of opposition Mr Sunak is likely to face when the bill is debated in the Lords.
But only one Conservative - the Earl of Dundee - rebelled against the government in Monday night's vote.
The vote was the first to be held under the Constitutional Reform and Governance Act 2010 - which set out new methods for the Lords to reject treaty ratification.
Once the Rwanda Bill is passed, the treaty will officially be ratified - though the government does not need the backing of parliament to approve the treaty.
The UK-Rwanda Treaty forms a central plank of the government's plan to deter migrants from crossing the Channel in small boats.
The treaty was drafted in response to the Supreme Court ruling that the Rwanda policy is unlawful.
Home Secretary James Cleverly signed the new legally binding treaty in Kigali, which he said would ensure people relocated to Rwanda are not at risk of being returned to a country where their life or freedom would be threatened - a process known as non-refoulement.
The treaty also establishes a new appeal body, which will be made up of judges with asylum expertise from a range of countries, to hear individual cases.
The government says Rwanda's asylum system would be monitored by an independent committee, whose powers to enforce the treaty would be beefed up.
The Monitoring Committee will develop a system which will enable relocated people and their lawyers to lodge complaints, the government says.
But the House of Lords backed a motion on Monday saying the treaty should wait until the government is able to prove Rwanda is safe.
The motion was proposed by a Labour former Attorney General, Lord Goldsmith, on behalf of the cross-party International Agreements Committee which listed 10 outstanding issues in the Rwandan asylum system.
The committee published a report last week that argued protections promised by the Rwanda Treaty, have not yet been introduced.
Lord Goldsmith said: "The government has already presented a Bill to Parliament asking it to make a judgment that Rwanda is safe now.
"And yet on the home secretary's own evidence, it cannot be so because the measures are not in place and have not been shown to be effective,
The Liberal Democrat spokesperson Lord Fox added: "There is more than enough reason to delay the ratification of the treaty until the conditions for its lawful operation are actually in place."
Although the vote is not binding on the government, it could feature in future legal challenges against the policy.
