Three pointings of the ROSAT PSPC have been used to map the X-ray emission of CTA 1. Emission is brightest at the center and extends in all directions to the boundaries of the remnant. The luminosity is model-dependent and (at 1.4 kpc), is, L(x)(0.1-2.4 keV) = 5-8 x 10(34) ergs s(-1). Emission from the center is harder and considerably brighter than that from the rim. Spectra from several parts of the remnant are fitted with two component models. Both a two-temperature thermal model and a model consisting of a power law and a soft thermal component produce good fits to all regions. The spectra do not allow us to distinguish between a remnant with hot evaporated material at the center, or an old Crab-like remnant. Both are plausible models for CTA 1. The ROSAT HRI was used to obtain an accurate location of a bright unresolved source within the remnant. This is identified as a background active galactic nucleus. Properties of eight weaker, unresolved X-ray sources within or nearby the remnant are tabulated