The government's withdrawal legislation had cleared parliament on Wednesday after peers, who had tried to secure additional rights including for unaccompanied child refugees, bowed to the will of MPs once the elected chamber overturned their demands.
The EU Parliament also must ratify the Brexit divorce deal before January 31, Britain's scheduled departure date.
Despite Johnson's repeated promise to "get Brexit done" on January 31, the day of departure only marks the start of the country's European Union exit.
Earlier on Thursday, Queen Elizabeth II signed the bill into law, finalizing preparations for the divorce on the British side after 10 months of extensions.
In the speech, he is due to call for national healing and unity, also making clear that after Brexit, the United Kingdom will expand a whole array of freedoms for Britons, including those related to fishing, immigration and trade.
Boris Johnson has said he will not take the option available to him in the withdrawal agreement of extending this transition period.
The formal talks are not expected to begin until March but the war of words is already intense.
Attention will now shift to a vote on the Brexit deal in the European Parliament next Wednesday.
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The UK's relationship with the bloc is set to remain unchanged during an 11 month transition period in which negotiators will thrash out details on the future relationship, including new agreements on security, trade and free movement.
The government's Brexit bill, which enshrines the agreement reached by Mr Johnson in October, is one step away from becoming law after completing its passage through Parliament without any changes.
Johnson returned as May's successor in July past year and has since managed to negotiate his own deal with Brussels and regain the government's control of parliament in a risky early election last month.
Britain and the European Union will then launch into negotiations on their future ties, racing to strike new relationships for trade, security and a host of other areas by the end of 2020.
So does everything change at 23:00 GMT on 31 January?
'At times it felt like we would never cross the Brexit finish line, but we've done it.
The Prime Minister, however, remains upbeat, insisting Britain can now move forward after years of wrangling over Brexit.
Johnson added, "Now we can put the rancor and division of the past three years behind us and focus on delivering a bright, exciting future, with better hospitals and schools, safer streets and opportunity spread to every corner of our country".