The short starts with a drunken stork delivering a baby to the nearest available home. Sylvester's wife, wanting a baby despite his objections, graciously receives the package. Sylvester is nonetheless excited- until he learns the baby is a mouse, at which point he tries to eat it. His wife quickly stops him, and when she goes out (and is not seen again afterwards), he tries again, only to stop after the mouse calls him "Daddy".
Sylvester's attitude changes entirely from this point on, and he takes him for a walk. Unfortunately, the neighborhood cats are not so enamored of the mouse, and he is forced to run back into the house. Several cats try to steal the mouse, using babysitter, salesman and Santa disguises, cutting holes in the floor, etc., only to be foiled by Sylvester, who for once, is on the winning end of the same traps that he usually ends up getting foiled by. The stork, meanwhile, returns (still drunk) to deliver the mouse to its proper home. Sylvester, believing it to be another cat, stops the mouse and is pulled up instead.
A later scene reveals two mice walking him (dressed as a baby) with the wife telling her husband, "Nothing like this ever happened on my side of the family!" and he looks at the audience in bewilderment as the cartoon irises out.
Streaming - Boomerang Streaming Service - Sylvester and Tweety section
The ABC version cuts all the scenes featuring the drunken stork delivering the mouse to Sylvester's house (and the part where Sylvester's wife suggests that they have a baby, then weeps and wishes she were dead, with Sylvester mocking her), creating a plot hole as to how the baby mouse ended up at their house.