After the Lords, the EU Withdrawal Bill will return to the House of Commons.
It would mean the government having to negotiate an global agreement enabling Britain to continue to participate in the European Economic Area (EEA) after exit day.
Last week, Brexit Secretary David Davis warned that amendments passed by Remainer peers could risk undermining negotiations with the EU. In total, they have voted through 12 amendments against government wishes.
Baroness Angela Smith, Labour's leader in the upper house, said the votes had afforded MPs a chance to consider the "finer details" of the Brexit bill.
He said: "On borders it would mean that we would have to continue to accept all four freedoms of the single market, including freedom of movement".
Some lawmakers criticized the government's plan to impose a specific date for Britain leaving the European Union, saying it would create significant difficulties if negotiations with Brussels went down to the wire.
"Peers have no faith in the British people to govern ourselves and prosper without the interference of the European Union in every aspect of our daily lives".
Labour MP Chuka Umunna, who had lobbied for peers to support all three amendments through his role with pro-Europe campaign group Open Britain, said: "This is a stunning victory for those who want to protect jobs and trade and keep our businesses linked to our biggest market".
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"It's for that reason that we need to ensure that any continuation of the customs union must include a continuation in the EEA or its equivalent".
The House of Lords backed a move to allow Britain's continued...
'We know the damage leaving the single market will do to our economy, to public services and to our NHS, so it would go against Labour's progressive values for the party not to vote in favour of these amendments tabled by Lord Alli, a leading equalities campaigner, in the Lords'.
Peers are seeking to amend the draft law to remove the fixed timing of Brexit at 11pm on March 29 next year, and to enable future governments to participate in European Union agencies after the United Kingdom leaves the bloc.
The duke said he was trying to help the government by giving ministers greater flexibility in negotiations and not to "thwart the process".
Lord Callanan said he saw "no reason" to vote for such a change.
Labour peers will today propose an amendment to key Brexit legislation so that March 29, 2019, is no longer fixed in law as the day of departure.
He told his fellow peers the amendment was "constructive" and aimed to "give some idea as to what sort of milk and honey might lie over the mountain once we have negotiated the wilderness journey".