Two soft X-ray images of the Chamaeleon I star-forming cloud obtained with the ROSAT Position Sensitive Proportional Counter are presented. Seventy reliable, and 19 possible additional, X-rav sources are found. Eighty percent of these sources arc certainly or probably identified with T Tauri stars formed in the cloud. Nineteen to 39 are proposed new ''weak'' T Tauri (WTT) stars which. when confirmed by optical spectroscopy, will significantly enlarge the known cloud population. Individual T Tauri X-ray luminosities range from approximately 6 x 10(28) to approximately 2 x 10(31) ergs s-1 (0.4-2.5 keV), or approximately 10(2)-10(4) times solar levels. The ROSAT images are an order of magnitude more sensitives, with 3 4 times more stellar identifications, than earlier Einstein Observatory images of the cloud. A wide range of issues is addressed by these data. The spatial distribution and Hertzsprung-Russell diagram locations of the stars indicate that WTT stars and ''classical'' T Tauri (CTT) stars are coeval. Their X-ray luminosity functions are also essentially identical, suggesting that CTT stars have the same surface magnetic activity as WTT stars. The X-ray luminosities of well-studied Chamaeleon I cloud members are strongly correlated with a complex of four stellar properties: bolometric luminosity, mass, radius, and effective temperature. The first relation can be expressed by the simple statement that low-mass Chamaeleon I stars have L(X)/L* = 1.6 x 10(-4), within a factor of +/-2 (1 sigma) and the radius relation by F(X) is-proportional-to R*. There is thus no evidence of magnetic saturation of the stellar surfaces. We find no evidence for the absorption of soft X-rays in CTT winds and/or boundary layers traced by the strength of the Halpha emission. The mean X-ray luminosity for an unbiased optically selected T Tauri sample is 1.6 x 10(29) ergs s-1, and we find evidence for temporal evolution of X-ray emission for stars within the pre-main-sequence evolutionary phase. The total pre-mainsequence population (M* > 0.1 M.) of the cloud is estimated to be greater than or similar to 200 stars, with X-ray-detected WTT stars outnumbering X-ray detected CTT stars by at least 2:1. The inferred star formation efficiency for the cloud cores is almost-equal-to 20%.