We present an improved version of the POTENT method for reconstructing the cosmological velocity and mass density fields from radial peculiar velocities, test it with mock catalogs, and apply it to the Mark III Catalog of Galaxy Peculiar Velocities. The method is improved in several ways: (1) the inhomogeneous Malmquist bias is reduced by grouping and corrected statistically in either forward or inverse analyses of inferred distances, (2) the smoothing into a radial velocity field is optimized such that window and sampling biases are reduced, (3) the density field is derived from the velocity field using an improved weakly nonlinear approximation in Eulerian space, and (4) the computational errors are made negligible compared to the other errors. The method is carefully tested and optimized using realistic mock catalogs based on an N-body simulation that mimics our cosmological neighborhood, and the remaining systematic and random errors are evaluated quantitatively. The Mark III catalog, with ~3300 grouped galaxies, allows a reliable reconstruction with fixed Gaussian smoothing of 10-12 h^-1 Mpc out to ~60 h^-1 Mpc and beyond in some directions. We present maps of the three-dimensional velocity and mass-density fields and the corresponding errors. The typical systematic and random errors in the density fluctuations inside 40 h^-1 Mpc are ±0.13 and ±0.18 (for Ω = 1). In its gross features, the recovered mass distribution resembles the galaxy distribution in redshift surveys and the mass distribution in a similar POTENT analysis of a complementary velocity catalog (SFI), including such features as the Great Attractor, Perseus-Pisces, and the large void in between. The reconstruction inside ~40 h^-1 Mpc is not affected much by a revised calibration of the distance indicators (VM2, tailored to match the velocities from the IRAS 1.2 Jy redshift survey). The volume-weighted bulk velocity within the sphere of radius 50 h^-1 Mpc about the Local Group is V₅₀ = 370 ± 110 km s^-1 (including systematic errors) and is shown to be mostly generated by external mass fluctuations. With the VM2 calibration, V₅₀ is in a similar direction and reduced to 305 ± 110 km s^-1.