Mr Blomfield, Labour MP for Sheffield Central, said: "Leaving the EU doesn't mean that we will have got Brexit done. We'll have completed the first step, departure, but the difficult stage is yet to come."
He added: "I would though, urge the Government, as we move into this next stage, not to over-interpret its mandate in the General Election.
Mr Blomfield said that while the Tories have "clearly secured an overwhelming majority of seats", most people "voted for parties that did not support the principle of getting Brexit done at any price".
He added that the Government "may find it hard to deliver on the high expectations that they have created over the last three-and-a-half years".
Stating that Labour will be voting against the Bill at third reading, Mr Blomfield said: "The decision of the general election isn't a mandate to bulldoze through a particular version at any cost on all the people of the United Kingdom, and the next few months need to be approached with sensitivity and with caution if we are to stay together as a United Kingdom."
It comes as MPs started debating the third reading of the European Union (Withdrawal Agreement) Bill, as it looks set to clear its final Commons hurdle this afternoon.
An amendment from the SNP has been selected by the Speaker. It calls for the Bill to be voted down because the Scottish Parliament withheld its consent to the Prime Minister's Brexit deal.
Opening the debate, Brexit minister James Duddridge said: "This Bill is essential in preparing our country for leaving the European Union and will ensure that the deal which has been reached can be implemented.
"It also ensures that we can protect the rights of citizens who have made their lives here and there is no hard border on the island of Ireland and that we take back control of money and our laws."