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Let $\cD$ be the Dirichlet space, namely the space of holomorphic functionson the unit disk whose derivative is square-integrable. We establish a newsufficient condition for a function $f\in\cD$ to be {\em cyclic}, i.e. for$\{pf: p\text{a polynomial}\}$ to be dense in $\cD$. This allows us to prove aspecial case of the conjecture of Brown and Shields that a function is cyclicin $\cD$ iff it is outer and its zero set (defined appropriately) is ofcapacity zero.

On the Brown--Shields conjecture for cyclicity in the Dirichlet space