  Let $\mathcal X$ be a regular variety, flat and proper over a complete
regular curve over a finite field, such that the generic fiber $X$ is smooth
and geometrically connected. We prove that the Brauer group of $\mathcal X$ is
finite if and only Tate's conjecture for divisors on $X$ holds and the
Tate-Shafarevich group of the Albanese variety of $X$ is finite, generalizing a
theorem of Artin and Grothendieck for surfaces to arbitrary relative dimension.
We also give a formula relating the orders of the group under the assumption
that they are finite, generalizing the formula given for a surface.
