The club said in a statement posted to its website on Tuesday afternoon: "It is with great sadness that the club can now confirm that Tom Baty, 26, from Durham City has died this afternoon following his admission to hospital on Sunday, 13 May".
Thomas Howard and Thomas Baty, of Durham, suffered difficulties breathing after returning from a nightclub and were taken to hospital in Colombo in the early hours of Sunday morning.
Mr Wilkinson also extended his best wishes to Mr Baty and those in Sri Lanka supporting him.
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The pair had been touring the country with Durham-based Clems Pirates RFC when they visited the club in Colombo.
Durham City RFC president Richard Wilkinson said: "On behalf of all members and friends of the DCRFC family we would like to extend our honest condolences to the family of Thomas Howard".
"We will also continue to support the Clems Sri Lankan Tour party by being in regular contact with senior club officials out in Sri Lanka".
"Thoughts and prayers at this hard time".
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The deceased player and other player were hospitalized by the hotel management as they were suffering from respiratory issues.
The team was staying in a hotel in Colombo after playing their opening match against Ceylonese Rugby and Football Club on Saturday.
Sri Lanka Police said a 26-year-old man had died in hospital at 12:00 local time on Sunday.
"We are assisting the family of a British man who has been hospitalised in Sri Lanka, and are in contact with the Sri Lankan medical services".
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Clems Pirates are described as a "fine example of veterans rugby", known in the northeast of England for "their sheer enthusiasm, terrific tours and their fundraising efforts".