In this paper, we give a Gr\"obner-Shirshov basis of the braid group
$B_{n+1}$ in Adyan-Thurston generators. We also deal with the braid group of
type $\bf{B}_{n}$. As results, we obtain a new algorithm for getting the
Adyan-Thurston normal form, and a new proof that the braid semigroup
$B^+_{n+1}$ is the subsemigroup in $B_{n+1}$.