Planar Turán number, denoted by ${\mathrm{e}\mathrm{x}}_{\mathcal{P}}(n,H)$, is the maximum number of edges in an $n$-vertex planar graph which does not contain $H$ as a subgraph. Ghosh, Győri, Paulos and Xiao initiated the topic of the planar Turán number for double stars. There were two double stars $S_{3,4}$ and $S_{3,5}$ that remained unknown. In this paper, we give the exact value of ${\mathrm{e}\mathrm{x}}_{\mathcal{P}}(n,S_{3,4})$.