Ann-Margrock (special guest star Ann-Margret, providing her own voice) comes to town for a big concert and stays with the Flintstones, who do not know who she is. After a few tries Fred and Barney both get on the Ann-Margrock Show.

Arnold the paper boy continues to annoy Fred as he is playing with Pebbles. He jokes that they will get married when they grow up. Fred then has a bad dream that Arnold grows up, becomes the local hero, takes his job at the quarry, his pride as a pool player and bowler, his friendship with Barney, and of course, marries Pebbles.

Fred kicks Barney and Betty out of the Flintstones' home and they want a baby badly and wish upon a falling star and the next morning a baby boy is found in a basket on their door step. He was named Bamm Bamm and he turns out to be the strongest baby in the world. Barney and Betty then take steps to legally adopt him. After almost being turned down, they are given the baby.