Preservation of Weyl invariance at the quantum level is a widely accepted consistency criterion for classically conformally invariant field theories. The paper determines the purely gravitational part of the Weyl anomalies in six dimensions by a purely cohomological method that identifies the anomalies with 1-cocycles of a certain coboundary operator. The cross-consistency conditions among diffeomorphism and Weyl anomalies are considered in detail. It is shown that these anomalies are in fact equivalent as they can be obtained from each other by adding a suitable local counterterm [due to W. A. Bardeen and B. Zumino, Nucl. Phys. B 244, 421-453 (1984)] to the action. By a simple two-dimensional example it is shown that the introduction of this counterterm corresponds to changing the regularization scheme.