The nearby barred spiral galaxy NGC 1313 has been observed with the PSPC instrument on board the ROSAT X-ray satellite. Ten individual sources are found. Three sources (X-1, X-2, and X-3 [SN 1978K]) are very bright (similar to 10(40) ergs s(-1)) and are unusual in that analogous objects do not exist in our Galaxy. We present an X-ray image of NGC 1313 and X-ray spectra for the three bright sources. The emission from the nuclear region (R less than or similar to 2 kpc) is dominated by source X-1, which is located similar to 1 kpc north of the photometric (and dynamical) center of NGC 1313. Optical, far-infrared, and radio images do not indicate the presence of an active galactic nucleus at that position; however, the compact nature of the X-ray source (X-1) suggests that it is an accretion-powered object with central mass M greater than or similar to 10(3) M.. Additional emission (L(x) similar to 10(39) ergs s(-1)) in the nuclear region extends out to similar to 2.6 kpc and roughly follows the spiral arms. This emission is from four sources with luminosity of several x 10(38) ergs s(-1) two of which are consistent with emission from Population I sources (e.g., supernova remnants, and hot interstellar gas which has been heated by supernova remnants). The other two sources could be emission from Population II sources (e.g., low-mass X-ray binaries). The bright sources X-2 and SN 1978K are positioned in the southern disk of NGC 1313. X-2 is variable and has no optical counterpart brighter than 20.8 mag (V-band). It is likely that it is an accretion-powered object in NGC 1313. The Type II supernova SN 1978K (Ryder et al. 1993) has become extraordinarily luminous in X-rays similar to 13 yr after optical maximum.