We provide sharp bounds for the isoperimetric constants of infinite plane graphs (tessellations) with bounded vertex and face degrees. For example, if $G$ is a plane graph satisfying the inequalities $p_1 \leq \mbox{deg}\ v \leq p_2$ for $v \in V(G)$ and $q_1 \leq \mbox{deg}\ f \leq q_2$ for $f \in F(G)$, where $p_1, p_2, q_1$, and $q_2$ are natural numbers such that $1/p_i + 1/q_i \leq 1/2$, $i=1,2$, then we show that \[ Φ(p_1, q_1) \leq \inf_S \frac{|\partial S|}{|V(S)|} \leq Φ(p_2, q_2), \] where the infimum is taken over all finite nonempty subgraphs $S \subset G$, $\partial S$ is the set of edges connecting $S$ to $G \setminus S$, and $Φ(p,q)$ is defined by \[ Φ(p, q) = (p-2) \sqrt{1 - \frac{4}{(p-2)(q-2)}}. \] For $p_1=3$ this gives an affirmative answer to a conjecture by Lawrencenko, Plummer, and Zha from 2002, and for general $p_i$ and $q_i$ our result fully resolves a question posed in the book by Lyons and Peres from 2016, where they extended the conjecture of Lawrencenko et al. to the above form.