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Which one is correct? a vote of thanks or the vote of thanks in 'It's my immense pleasure to propose a vote of thanks/ the vote of the thanks at the end of this programme'.

"a vote of thanks"

Definitely a "first mention" context. Even any subsequent referring back probably wouldn't involve 'the', so as to avoid repetition (A: There was a show of hands. B: How did it/?the show of hands/the vote [note the noun in that last option has changed to a synonym providing variety] go? What was decided?).

I propose a vote of thanks to Lorikeet and Fluffyhamaster.