Let $R>0$ and let $f$ be holomorphic at $0$ and meromorphic in $\{z\in\mathbb{C}:|z|<R\}$. Assume that $f$ has exactly $n$ poles in this disk, counted with multiplicity, and no other singularities in the disk. No hypothesis is imposed on the boundary circle $|z|=R$. For all sufficiently large $m$, assume that the type $(m,n)$ Pade entry is regular, so there are uniquely determined polynomials $p_{m,n}$ and $q_{m,n}$ with $\deg p_{m,n}\leq m$, $\deg q_{m,n}\leq n$, and $q_{m,n}(0)=1$, giving the rational approximant $r_{m,n}=p_{m,n}/q_{m,n}$. Then, as $m\to\infty$ through these regular entries, the denominators $q_{m,n}$ converge coefficientwise to the normalised polynomial whose zeros are the poles of $f$ in $\{z\in\mathbb{C}:|z|<R\}$, counted with multiplicity. Moreover $r_{m,n}\to f$ uniformly on compact subsets of $\{z\in\mathbb{C}:|z|<R\}$ that avoid the poles of $f$.