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Measurement of the amount of $^{44}$Ti ejected by a supernova is essential for confirming the accuracy of nucleosynthesis calculations of this isotope production in incomplete Si burning during the evolution of a massive star, during explosive Si burning, or during an alpha-rich (low-density) freeze-out of nuclear statistical equilibrium. The recent CGRO/COMPTEL discovery of the 1.16 MeV $\gamma$-ray line of $^{44}$Ti from the Cas A supernova remnant (SNR) is highly significant because it implies a relatively high yield of $^{44}$Ti ($\sim$ 2--3 $\times$ 10$^{-4}$ M$_{\sun}$) compared with explosive nucleosynthesis calculations ($\sim$ 1 $\times$ 10$^{-4}$ M$_{\sun}$) and because it demonstrates that unobserved supernovae in the galaxy from the past several hundred years are possibly detectable through their $^{44}$Ti $\gamma$-line emission with CGRO and future missions. Cas A, the youngest known supernova remnant in the Galaxy and a strong radio and X-ray source, was observed by CGRO/OSSE July 16 - August 6, 1992. Its proximity($\sim$ 3 kpc) and its young age ($\sim$ 300 yrs) make Cas A the best candidate among known supernova remnants for detecting $^{44}$Ti $\gamma$-ray lines. In contrast to the COMPTEL result, we find no evidence for $\gamma$-radiation at 67.9 keV, 78.4 keV, or 1.157 MeV, the three strongest $^{44}$Ti $\gamma$-ray lines. From simultaneous fits to the three lines our 99\% confidence upper limit of the $\gamma$-line fluxes is 5.2$\times$10$^{-5} \; \gamma \;$ cm$^{-2} \;$ s$^{-1}$ for each line feature. We also report upper limits for 4.44 MeV $^{12}$C nuclear excitation line, which could be caused by interactions of accelerated particles in the supernova remnant, and for the hard X-ray continuum.
