  We establish a linear relation between the $a$-type Weyl anomaly and the 't
Hooft anomaly coefficients for the $R$-symmetry and gravitational anomalies in
six-dimensional $(1,0)$ superconformal field theories. For RG flows onto the
tensor branch, where conformal symmetry is spontaneously broken, supersymmetry
relates the anomaly mismatch $\Delta a$ to the square of a four-derivative
interaction for the dilaton. This establishes the $a$-theorem for all such
flows. The four-derivative dilaton interaction is in turn related to the
Green-Schwarz-like terms that are needed to match the 't Hooft anomalies on the
tensor branch, thus fixing their relation to $\Delta a$. We use our formula to
obtain exact expressions for the $a$-anomaly of $N$ small $E_8$ instantons, as
well as $N$ M5-branes probing an orbifold singularity, and verify the
$a$-theorem for RG flows onto their Higgs branches. We also discuss aspects of
supersymmetric RG flows that terminate in scale but not conformally invariant
theories with massless gauge fields.
