He was born near Beroun, in the present-day Czech Republic. He attended a Jesuit school in Příbram, and studied organ and composition with Josef Seger. He was ordained as priest in 1786.Wolfgang Amadeus Mozart met Gelinek while in Prague for a performance of his opera Don Giovanni (first performed there in 1787); through his recommendation, Gelinek was appointed chaplain and keyboard teacher at the court of Count Kinsky. After about two years, Gelinek accompanied the Count's family to Vienna, where he was in the service of Prince Joseph Kinsky. He stayed there for 13 years.

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