Two-dimensional spectroscopy of the low-redshift active galaxy NGC 4303 obtained with the Two-Dimensional Fiber ISIS System is presented. The ionization structure and velocity field of the nuclear region of this galaxy, where a compact UV-bright spiral structure connected with the UV-bright core has been detected with the Hubble Space Telescope (HST; Colina et al. 1997a), are investigated in detail. The ionized gas shows a structure similar to that observed in the high spatial resolution HST UV-continuum image. The Hβ and Hα emission is dominated by the nuclear star-forming regions, while the core of the galaxy is the brightest region in the [O III] lambda5007 and [N II] lambda6584 emission lines. The optical emission line ratios of the nuclear star-forming regions are characteristic of young (ages 2-3 Myr) star-forming regions, while the core of the galaxy shows the emission-line ratios of a low-luminosity AGN in between an [O I]-weak LINER and a low-excitation Seyfert 2. The nature of the ionizing source located at the core of the galaxy is still unsolved. Its luminosity and optical emission line ratios are compatible with the presence of a young (~3.5 Myr) massive (~8x10^4 M_solar) cluster of stars, but are also consistent with the existence of a power-law nonthermal ionizing source. Whatever the nature of this ionizing source, the nuclear star-forming spiral dominates the ionizing radiation, contributing about 90% of the total ionizing flux. The velocity field of the ionized gas, as measured by the Hβ emission line, is consistent with that of a massive rotating disk characterized by a radius of 300 pc, a rotation velocity of 85 km s^-1, an inclination of 45 deg with respect to the line of sight, and a kinematic major axis oriented along P.A. 130 deg. The inferred dynamical mass inside a radius of 300 pc is 5.0x10^8 M_solar. The mass and size of this nuclear rotating disk is similar to those recently detected in ultraluminous infrared galaxies like Mrk 231 and Arp 220. The observed line profiles of the high-excitation [O III]-emitting gas show the presence of two kinematically distinct gaseous components. The main component follows the velocity pattern of the massive rotating disk. The secondary component has an amplitude of +/-350 km s^-1, and a minor axis oriented, in projection, almost perpendicular to the minor axis of the low-ionization gas. This second[O III] velocity component could represent gas located in a nuclear ionizing cone, i.e., gas flowing outward outside the plane of the low-excitation gas disk, and being ionized by the UV-bright source located at the core of the galaxy. The proximity of NGC 4303 together with the detection of (1) an AGN-like nucleus, (2) a compact nuclear star-forming spiral structure connected to the nucleus, (3) a massive rotating nuclear disk, and (4) radially flowing high-excitation gas makes this galaxy an ideal candidate for the study of the fueling of active galaxies and the starburst-AGN connection.