The president presented the legislation to Sejm (lower house) Speaker Elzbieta Witek at a ceremony held at pl. Pilsudski, where the palace once stood.
The Saski Palace was one of the most distinctive buildings in pre-war Warsaw before its destruction. Now only a fragment of the palace, which houses the grave of the Unknown Solider, remains.
President Duda said that the ceremony was a very special moment not only for Warsaw, but also for the entire country.
"Unfortunately, during the course of this reconstruction, which took place decades ago, there wasn't enough strength and will to rebuild the entire frontage of the Saski Palace," he said.
The president also pointed out that the reconstruction of the palace was a huge financial undertaking, but it would "crown the entire... rebuilding of the capital."
"I believe that today's Republic, which is sovereign again, fully independent again, and gaining affluence... will complete this work," he added.
According to a report by the Gazeta Wyborcza daily, the entire project could cost PLN 1.5 billion (EUR 331 million).
Upon receiving the draft act Witek expressed her hope that it would be passed by both the upper and lower houses of parliament as quickly as possible.
She said that she would ensure that "the Sejm will deal immediately with the processing of this law."
Prime Minister Mateusz Morawiecki called for the reconstruction of the Saski Palace to bring together political foes.
"Let it be a nationwide, non-partisan project, a project that will bring together all Poles," he said.
He added that the palace had a chance to become the "beating heart of the III Republic of Poland." (PAP)