The restoration bill states that the reconstruction must create a cathedral that is faithful to the "last known visual state" of the building, reports French newspaper Le Monde.
It also says that the restoration must recreate Eugène Viollet-le-Duc's spire, and that any use of new materials will need to be justified.
The French Senate – the upper house of the French parliament – adopted the text late on Monday night. Senators added the clauses to the bill created by the National Assembly, the French parliament's lower house.
Députés of the National Assembly and senators both have to agree on the text before the bill can move forward. Once passed by both houses, it will determine how the restoration of the state-owned building proceeds.
Bill would end speculation
If the bill progresses in its current state, it would end the speculation over how the cathedral will be rebuilt.
It would seem to go against the desires of both France's president, Emmanuel Macron, and prime minister, Edouard Philippe.
Macron has called for "an inventive reconstruction", and following the fire, Philippe announced an international competition to design a new spire for the cathedral. He told reporters: "As is often the case in the evolution of heritage, we should endow Notre-Dame with a new spire."
These announcements have led to numerous architects and designers coming up with inventive proposals for how the cathedral could be rebuilt with a modern twist.
Several designers, including Vincent Callebaut, Miysis Studio and StudioNAB, have proposed adding a glass roof to the building, while numerous others have proposed alternative spires to top the building.
Among the more outrageous proposals include turning Notre-Dame into a car park, a swimming pool or a McDonald's.
Rebuild to follow standard building laws
When passed into law, the bill will establish a framework of how the restoration will be undertaken.
The senate's agreed text that would establish a public body to look after the restoration. This would be overseen by the Ministry of Culture.
However, it removed text from the bill that would have allowed this body to avoid standard planning, environmental and heritage regulations.
If agreed by the National Assembly, this would mean the project would have to adhere to all standard building laws, potentially impacting Macron's commitment to rebuild the cathedral before the 2024 Olympics in Paris.
This timescale has already been questioned by heritage experts, who have urged the president to abandon this ambitious restoration deadline.
Notre-Dame, on the Ile de la Cité in central Paris, is a cathedral that was largely built in the 13th century, before undergoing a substantial restoration in the 19th century. A large fire on the 15th April destroyed much of the landmark's original oak roof and the spire later by Viollet-le-Duc.
Main image is by Clem.