  In a search for $\omega$ mesic states, the production of $\omega$-mesons in
coincidence with forward going protons has been studied in photon induced
reactions on $^{12}$C for incident photon energies of 1250 - 3100 MeV. The
$\pi^0 \gamma$ pairs from decays of bound or quasi-free $\omega$-mesons have
been measured with the CBELSA/TAPS detector system in coincidence with protons
registered in the MiniTAPS forward array. Structures in the total energy
distribution of the $\pi^0 \gamma$ pairs, which would indicate the population
and decay of bound $\omega~^{11}$B states, are not observed. The $\pi^0 \gamma$
cross section of 0.3 nb/MeV/sr observed in the bound state energy regime
between -100 and 0 MeV may be accounted for by yield leaking into the bound
state regime because of the large in-medium width of the $\omega$-meson. A
comparison of the measured total energy distribution with calculations suggests
the real part $V_0$ of the $\omega~^{11}$B potential to be small and only
weakly attractive with $V_0(\rho=\rho_0) = -15\pm$ 35(stat) $\pm$20(syst) MeV
in contrast to some theoretical predictions of attractive potentials with a
depth of 100 - 150 MeV.
