Weck studied acting at the University of Music and Performing Arts Vienna and the Max Reinhardt Seminar.

He made his film debut in 1954 and had a supporting role as Karl Ludwig of Austria in the 1955 historical drama Sissi (1955) with Romy Schneider.

Weck found himself typecast in comedic and romantic roles in light entertainment films.

He was also successful as a stage director for a number of theatres which work together in the organization Vereinigte Bühnen Wien, co-founded by Weck in 1987.

As a Theater manager in Vienna during the 1980s, he was responsible for the German-language premieres of the Andrew Lloyd Webber musicals Cats and The Phantom of the Opera.