  A systematic study of the inclusive ($0\nu+2\nu$) double beta ($\beta\beta$)
decay of $^{100}$Mo to various excited final states of $^{100}$Ru was
performed. Utilizing two large HPGe detectors operated in coincidence, a search
for the subsequent deexcitation gamma-gamma cascades was conducted. A 1.05-kg
sample of isotopically enriched (98.4%) $^{100}$Mo was investigated for 455
days, yielding an unambiguous observation of the double beta decay of
$^{100}$Mo to the $0_1^+$ state (1130.3 keV) of $^{100}$Ru. This excited final
state decays via the $0_1^+ \to 2_1^+ \to 0_{gs}^+$ sequence (with
$E_{\gamma1}$ = 590.8 keV and $E_{\gamma2}$ = 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
$\beta\beta($0^+ rightarrow 0_1^+)$ transition of $T_{1/2}^{(0\nu+2\nu)} = [
6.0^{+1.9}_{-1.1} (\rm{stat}) \pm 0.6(\rm{syst})] \times 10^{20}$ years. Lower
limits on decay half-lives were achieved for higher excited final states.
