Police said another player is receiving treatment.
Baty was transferred to the private Nawaloka hospital where he too died on Tuesday afternoon despite intensive care, officials said.
The Durham City Clem's Pirates arrived Sri Lanka on Thursday (10) last week with a schedule of a few friendly matches and coaching camps in Colombo and a few major cities of the country.
Mr Baty's teammate, Thomas Howard, 26, died on Sunday - also as a result of breathing difficulties.
Several players had gone to the nightclub after playing their tour-opener against Ceylonese Rugby and Football Club in Colombo on Saturday.
The death was confirmed to News 1st, by Police Media Spokesperson SP Ruwan Gunasekara.
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Durham City Rugby Football Club of amateur player Tom Baty, who remains in hospital in the city of Colombo in Sri Lanka after suffering breathing problems on Sunday.
In a statement released on Monday following the announcement of Mr Howard's death, Durham City RFC president Richard Wilkinson said: 'Both players and Clems Pirates hold a special place in our rugby family, and our continued support is unconditional.
She added: "We are assisting the family of a British man who has been hospitalized in Sri Lanka, and are in contact with the Sri Lankan medical services".
The pair's breathing issues were not rugby-related, the club said, but no further details of the fatal incident were provided.
'My thoughts are with the families and friends and team-mates, ' the councillor said.
Neil Foster, chief executive of Tynedale RFC wrote: "Tragic news".