  Two neutrino double beta decay of $^{76}$Ge to excited states of $^{76}$Se
has been studied using data from Phase I of the GERDA experiment. An array
composed of up to 14 germanium detectors including detectors that have been
isotopically enriched in $^{76}$Ge was deployed in liquid argon. The analysis
of various possible transitions to excited final states is based on coincidence
events between pairs of detectors where a de-excitation $\gamma$ ray is
detected in one detector and the two electrons in the other.
  No signal has been observed and an event counting profile likelihood analysis
has been used to determine Frequentist 90\,\% C.L. bounds for three
transitions: ${0^+_{\rm g.s.}-2^+_1}$: $T^{2\nu}_{1/2}>$1.6$\cdot10^{23}$ yr,
${0^+_{\rm g.s.}-0^+_1}$: $T^{2\nu}_{1/2}>$3.7$\cdot10^{23}$ yr and ${0^+_{\rm
g.s.}-2^+_2}$: $T^{2\nu}_{1/2}>$2.3$\cdot10^{23}$ yr. These bounds are more
than two orders of magnitude larger than those reported previously. Bayesian
90\,\% credibility bounds were extracted and used to exclude several models for
the ${0^+_{\rm g.s.}-0^+_1}$ transition.
