Uniform boundedness for rational points

We extend an earlier result by Dan Abramovich, showing that a conjecture of S. Lang's implies the existence of a uniform bound on the number of $K$-rational points over all smooth curves of genus $g$ defined over $K$, where $K$ is any number field of fixed degree $d$, and $g$ is an integer greater than 1. The bound depends only on the genus $g$ and the degree of the number field $K$.