Agreement negotiated with European Union ensures that U.K. will leave at end of month.

The Brexit divorce deal completed its passage through Parliament on Wednesday evening, ensuring that the U.K. will leave the European Union at the end of the month.

After passing the upper and lower parliamentary chambers, the Brexit divorce deal that Prime Minister Boris Johnson negotiated with the EU last year now just needs signoff from the queen to become British law.

On the European side, the EU council and commission presidents are due to sign off on the agreement Friday. The European Parliament needs to vote it through next Wednesday, but that step is considered a formality.

The passage of the bill through Parliament ends nearly four years of British political paralysis sparked by the vote to leave the EU in 2016. “At times it felt like we would never cross the Brexit finish line, but we’ve done it,” Mr. Johnson said.

Mr. Johnson’s commanding majority in Parliament following his December election victory meant that the Brexit deal went through Parliament un-amended. It paves the way for the U.K. to end its 47-year relationship with the EU on the evening of Jan. 31.

The divorce deal covers payments owed to the EU and citizens’ rights after Brexit, as well as a mechanism to avoid a hard border on the island of Ireland.

With the terms of the divorce with the EU all but settled, attention has turned to the nature of the two sides’ future relations. After Jan. 31, the U.K. will keep its practical ties to the EU until at least the end of 2020, while it negotiates new deals to cover their future relationship.