Publix Super Market will close and rebuild store in Venice VENICE — The Publix Super Market in Venice Village Shoppes at 4173 Tamiami Trail S. will close on Saturday and ultimately will be replaced with a new Publix on that same site. “All of the associates have been assigned to new locations in the area and the new store did not prompt the closing of that store,” said Publix community relations manager Brian West, referencing the Publix that opened at West Villages MarketPlace. “A lot of times we get concerns from our customers about the associates that work in the store when we close it.”