Legislators from all but one party backed in the first reading a bill that would make Cavallier a Dane.
The left-wing, anti-royalist Red-Green Alliance said it would abstain from the final vote, but all other parties supported the measure, meaning the vote is now considered a formality.
The wedding will be held at the Moegeltoender church in southern Denmark, just north of the German border.
Foreigners marrying into Denmark's royal family become Danish citizens. Recent examples are Joachim's former wife, Hong Kong-born Alexandra Manley, and Crown Prince Frederik's wife, Mary, who was born an Australian citizen.
In October, Joachim, 38, announced he was engaged to Cavallier.
He divorced Princess Alexandra in 2005, after 10 years of marriage. The divorce was the first in Europe's oldest monarchy in nearly 160 years. She has since remarried, giving up her Danish title.
The couple has two sons, Prince Nikolai, 8, and Prince Felix, 5.
Joachim is fourth in line to Denmark's throne.