A systematic study of the inclusive ($0\ensuremath{\nu}+2\ensuremath{\nu}$) double beta ($\ensuremath{\beta}\ensuremath{\beta}$) decay of $^{100}\mathrm{Mo}$ to various excited final states of $^{100}\mathrm{Ru}$ was performed. Utilizing two large HPGe detectors operated in coincidence, a search for the subsequent deexcitation $\ensuremath{\gamma}\ensuremath{\gamma}$ cascades was conducted. A 1.05-kg sample of isotopically enriched $(98.4%)$$^{100}\mathrm{Mo}$ was investigated for 455 days, yielding an unambiguous observation of the $\ensuremath{\beta}\ensuremath{\beta}$ decay of $^{100}\mathrm{Mo}$ to the ${0}_{1}^{+}$ state (1130.3 keV) of $^{100}\mathrm{Ru}$. This excited final state decays via the ${0}_{1}^{+}\ensuremath{\rightarrow}{2}_{1}^{+}\ensuremath{\rightarrow}{0}_{\mathrm{gs}}^{+}$ sequence (with ${E}_{\ensuremath{\gamma}1}=590.8$ keV and ${E}_{\ensuremath{\gamma}2}=539.5$ keV), and 22 such coincidence events were detected, with a continuous background estimated to be 2.5 events. This counting rate corresponds to a decay half-life for the $\ensuremath{\beta}\ensuremath{\beta}({0}^{+}\ensuremath{\rightarrow}{0}_{1}^{+})$ transition of ${T}_{1/2}^{(0\ensuremath{\nu}+2\ensuremath{\nu})}=[6.{0}_{\ensuremath{-}1.1}^{+1.9}(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.6(\mathrm{syst})]\ifmmode\times\else\texttimes\fi{}{10}^{20}$ years. Lower limits on decay half-lives were achieved for higher excited final states.