An exact sequence relating Br (X), the Brauer group of a regular scheme of dimension≦ 2, and Amitsur cohomology (obtained as the cohomology of the sheaf of units on an appropriate Grothendieck topology) is derived by functorial methods. In order to do this we first show that any torsion element of H 1 (X et, G m), ie, Pic (X), and H 2 (X et, G m), ie, Br (X), is split by a finite, faithfully flat covering Y→ X. After proving a divisibility result for Pic (X) under such coverings and some preliminary investigation of cohomology in the topology defined from such coverings, the exact sequence which is analogous to that of Chase and Rosenberg is obtained.