We present BeppoSAX follow-up observations of GRB 980425 obtained with the Narrow Field Instruments (NFI) in 1998 April, May, and November. The first NFI observation has detected within the 8' radius error box of the gamma-ray burst (GRB) an X-ray source positionally consistent with the supernova 1998bw, which exploded within a day of GRB 980425, and a fainter X-ray source, not consistent with the position of the supernova. The former source is detected in the following NFI pointings and exhibits a decline of a factor of 2 in six months. If it is associated with SN 1998bw, this is the first detection of X-ray emission from a Type I supernova above 2 keV. The latter source exhibits only marginally significant variability. The X-ray spectra and variability of the supernova are compared with thermal and nonthermal models of supernova high-energy emission. Based on the BeppoSAX data, it is not possible to establish firmly which of the two detected sources is the GRB X-ray counterpart, although probability considerations favor the supernova.