This paper presents new X-ray observations of Mrk 231, an active galaxy of particular interest because of its large infrared luminosity and the presence of several blueshifted broad absorption line (BAL) systems, a phenomenon observed in a small fraction of QSOs. A ROSAT HRI image of Mrk 231 is presented, which shows an extended region of soft X-ray emission covering several tens of kiloparsecs, consistent with the extent of the host galaxy. An ASCA observation of Mrk 231 is also presented. Hard X-rays are detected, but the data show no significant variability in X-ray flux. The hard X-ray continuum is heavily attenuated, and X-ray column estimates range from similar to 2 x 10(22) to 10(23) cm(-2), depending on whether the material is assumed to be neutral or ionized, and depending on the model assumed for the extended X-ray component. These ASCA data provide only the second hard X-ray spectrum of a BAL active galactic nucleus presented to date. The broadband spectral energy distribution of the source is discussed. While Mrk 231 is X-ray weak compared to Seyfert 1 galaxies, it has an optical-to-X-ray spectrum typical of a QSO.