Using scanning tunneling spectroscopy, we investigated the temperature dependence of the quasiparticle density of states of overdoped ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CuO}}_{6+\ensuremath{\delta}}$ between 275 mK and 82 K. Below ${T}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}10\phantom{\rule{0ex}{0ex}}\mathrm{K}$, the spectra show a gap with well-defined coherence peaks at $\ifmmode\pm\else\textpm\fi{}{\ensuremath{\Delta}}_{p}\ensuremath{\simeq}12\mathrm{meV}$, which disappear at ${T}_{c}$. Above ${T}_{c}$, the spectra display a clear pseudogap of the same magnitude, gradually filling up and vanishing at ${T}^{*}\ensuremath{\simeq}68\mathrm{K}$. The comparison with ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8+\ensuremath{\delta}}$ demonstrates that the pseudogap and the superconducting gap scale with each other, providing strong evidence that they have a common origin.