On May 15, 1973 a balloon-borne gamma-ray telescope belonging to the Rice University Gamma-Ray Astronomy Group was launched from Palestine, Texas. Both source and background segment counting rates were measured over the 56 keV to 12.4 MeV energy range from 118 CDT to 1313 CDT. In each of the source segments, the detector was pointed at the Seyfert galaxy NGC 1275 in the Perseus cluster. However the entire cluster was within the field of view of the detector during the observations of this galaxy. The experiment was carried out at altitudes corresponding to a residual atmospheric depth of approximately 4 gm/cm. The detector consisted of a central crystal of Nal (Tl) which was actively collimated by a shield of the same material. The instrument had a beam width of 13° full width half maximum. The results obtained show that NGC 1275 in particular and the Perseus cluster in general do not constitute a detectable gamma-ray source. A series of upper limits to the flux in the 56 keV to 12.4 MeV energy range was computed.