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18 year old Signaller Tyler Pearson, of 2 Signal Regiment, has not been seen since entering the water.
Tyler's family have been informed and are being supported by specially trained police and army officers.
North Yorkshire Police were called at 2.28am on Thursday 3 April 2014, after a member of the public reported seeing a man in the water near Ouse Bridge.
Officers attended the scene and with the assistance of the Fire and Rescue Service began a search of the river and the river banks.
Searches are continuing today in and around the River Ouse, using numerous police resources including the Regional Underwater Search Unit and other specialist search teams.
Superintendant Phil Cain, of York Safer Neighbourhood Command, said:
"It is believed that Tyler entered the river voluntarily during a night out with a group of friends but at this time we cannot confirm whether he has exited the water safely.
"He was seen to cross Ouse Bridge with two other people and then enter the river near the Lowther public house on King's Staith.
"It is believed that he was attempting to swim across the river to Queen's Staith.
"It is important that we establish the exact circumstances surrounding this incident and I am appealing to anyone who saw Tyler either enter or exit the river to get in touch with North Yorkshire Police as soon as possible."
Steve Waddington, Assistant Director of Housing and Community Safety at City of York Council, said:
"Our thoughts are with those affected by recent tragic events. We continue to work with agencies and voluntary organisations to raise awareness of the need to take care by the city's rivers and are determined to prevent river-related incidents where we can.
"As a partner of the River Ouse Safety Advisory Group, we work together to provide support and safety advice through the joint emergency plan which we maintain, develop and exercise.
"We urge everyone who is out and about in the city centre to watch out for each other and stay together when heading home so that should an accident occur, there is someone to raise the alarm."