The nearby barred spiral galaxy NGC 1313 has been observed with the PSPC instr- ument 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 (~10^40 erg/s) and are unusual in that analogous objects do not exist in our Galaxy. We present an X-ray image of NGC~1313 and \xray spectra for the three bright sources. The emission from the nuclear region (R ~< 2 kpc) is dominated by source X-1, which is located ~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 \xray source (X-1) suggests that it is an accretion-powered object with central mass M >~ 10^3 Msun. Additional emission (L_X ~ 10^39 erg/s) in the nuclear region extends out to ~2.6 kpc and roughly follows the spiral arms. This emission is from 4 sources with luminosity of several x 10^38 erg/s, 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 \xray 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 \etal 1993) has become extra- ordinarily luminous in X-rays $\sim$13 years after optical maximum.