Brexit: MPs back "alternative arrangements" for Irish backstop
The proposal - put forward by Tory MP Sir Graham Brady - had the support of the government and won by 16 votes.
Theresa May had urged MPs to vote in favour of it, to give her a mandate to return to Brussels and re-open negotiations in order to secure a "legally binding change".
But the EU has said it will not change the legal text agreed with the UK PM.
MPs voted on a string of amendments to Mrs May's plan to change the direction of Brexit.
Read more: May set for further talks with EU leaders