Vale have dropped to 15th in League Two, 12 points from the play-offs.
The club are currently under a transfer embargo and staff and players have only been paid 25% of their wages.
At the hearing, Judge Barker QC said: "In some ways this is a very sad matter.
"The sale of a football club is never easy. Hopefully when it rises, it will flourish."
In a statement released after the hearing, Stoke-on-Trent City Council said: "Mohammed Pervez, leader of Stoke-on-Trent City Council, has today confirmed that the High Court in Birmingham has granted the Council's application to place Port Vale (Valiant 2001) Football Club Limited in administration.
"As previously stated the administrators will be Begbies Traynor."
Speaking on Tuesday, Pervez said: "We have given a clear steer to officers to explore putting Port Vale into administration.
Board not to blame - Deakin
"This is as opposed to liquidation, which would see the end of the club."
The administrators will look to find a buyer to keep the club alive and they may turn to Mo Chaudry, who had an offer of £1.2m for 54% of shares turned down last summer and is watching developments at the club closely.
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