The regionalised draw was made within the last few minutes at St Mirren Park in Paisley, with the Greenock men in the south section.
It paired Jim Duffy's men with their near rivals from the other side of the River Clyde, who they will also face four times on league duty this term.
The Sons are under new management after ex-Ton midfielder Stevie Aitken took over the hotseat from Ian Murray this summer.
The tie is scheduled to be played on Saturday 25 July.
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