Calculating the quasiparticle (QP) band structure of two-dimensional (2D) materials within the GW self-energy approximation has proven to be a rather demanding computational task. The main reason is the strong $q$ dependence of the 2D dielectric function around $q=0$ that calls for a much denser sampling of the Brillouin zone (BZ) than is necessary for similar three-dimensional solids. Here, we use an analytical expression for the small $q$ limit of the 2D response function to perform the BZ integral over the critical region around $q=0$. This drastically reduces the requirements on the $q$-point mesh and implies a significant computational speedup. For example, in the case of monolayer ${\mathrm{MoS}}_{2}$, convergence of the ${\text{G}}_{0}{\text{W}}_{0}$ band gap to within $\ensuremath{\sim}0.1$ eV is achieved with $12\ifmmode\times\else\texttimes\fi{}12 q$ points rather than the $36\ifmmode\times\else\texttimes\fi{}36$ mesh required with discrete BZ sampling techniques. We perform a critical assessment of the band gap of the three prototypical 2D semiconductors, ${\mathrm{MoS}}_{2}, h$-BN, and phosphorene, including the effect of self-consistency at the ${\mathrm{GW}}_{0}$ level. The method is implemented in the open source code gpaw.