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.

All 33 crew died when the El Faro disappeared sailing from Florida to Puerto Rico in October 2015.
Its wreckage was discovered a month later but search crews were not able to retrieve the black box until now.
It is hoped the data will help determine what happened in the final hours before the boat's sinking.
"The recovery of the recorder has the potential to give our investigators greater insight into the incredible challenges that the El Faro crew faced,'' said Christophe Hart, chairman of the National Transportation Safety Board (NTSB).
The black box was found in April this year but only recovered after search crews drove a remotely operated vehicle down to the ship's resting place to remove it.
Hearings earlier this year looked at Captain Michael Davidson's decision to leave port with a huge storm brewing, and the safety record of the ship's owner, Tote Services.
A panel found that Davidson was aware of the hurricane and had planned a safer route but may have had out-of-date weather advice.
The El Faro sent out a distress signal on 1 October, saying it had lost power and was taking on water
The ship was eventually found 15,000 ft (4,570m) under the sea. A body wearing a survival suit was recovered shortly after the sinking, as was a heavily damaged lifeboat with no one aboard.

The seats in the Grand Tier are located five boxes away from the royal box in the Grade I listed building.
Agents Harrods Estates said it was the first time such a box had come on the market in almost a decade and there had "already been a bit of interest".
Buying the box will give the owner access to two-thirds of events.
They will also become a member of the Corporation of the Hall of Arts and Science which helps to operate the South Kensington building.
Nicholas Shaw of Harrods Estates said it was "the most expensive box we have ever offered for sale".
The box is on the western side of the auditorium and has approximately 849 years remaining on its lease.
A five-seat box situated on the second tier, whose previous owners included Earl Spencer and the Duke of Devonshire, went on sale in 2011 for £550,000.
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.
[NEXT_CONCEPT]
Search teams have recovered the black box data recorder from a cargo ship that sank last year near the Bahamas during Hurricane Joaquin.
[NEXT_CONCEPT]
A 12-seat private box in the Royal Albert Hall close to where the Queen sits when she attends events has gone on sale for £2.5m.