If finding enough cardboard to use as a weed barrier under mulch is a problem, I've found shredded office waste paper works as well if you soak it in a bucket of water and then put a layer of an inch or so down, topped with some grass clippings or compost. Probably best to avoid coloured inks or glossy finishes though.

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All at once I hear your voice
And time just slips away
Bonnie Rait