It will determine whether irregularities were committed and the election should be held again.
Freedom Party candidate Norbert Hofer narrowly lost to the former leader of the Greens, Alexander Van der Bellen, by less than a percentage point.
The court heard from about 90 witnesses during two weeks of public hearings.
The Freedom Party argues that the way postal votes were handled was among numerous irregularities that culminated in its defeat to the Greens by just under 31,000 votes.
It says that postal ballots were illegally handled in 94 out of 117 districts.
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If elected Mr Hofer would have been the first far-right head of state of an EU country,
Preliminary results in the vote count suggested he had a narrow lead but that disappeared after about 700,000 postal votes were counted.
The party also claims it has evidence that under-16s and foreigners were allowed to vote.
Mr Van der Bellen was eventually declared the winner of the largely ceremonial post with 50.3% of the vote against Mr Hofer's 49.7%. The president elect is due to be sworn in on 8 July.
His lawyer insisted that any irregularities that did exist had a negligible impact on the ballot.
Correspondents say that if the Constitutional Court disagrees with him, the election would be invalidated - a move that would send shockwaves throughout the country.
Such a ruling would also mean that Austria would be compelled to hold another vote in the coming months.
If a new election is ordered, departing President Heinz Fischer will be replaced on a temporary basis by three parliamentary officials, including Mr Hofer.
Austria's Constitutional Court is due to rule on the far-right Freedom Party's challenge to the result of last month's presidential run-off vote.