This work builds on and extends our previous effort (Tsyganenko et al., 2003) to develop a dynamical model of the storm-time geomagnetic field in the inner magnetosphere, using space magnetometer data taken during 37 major events in 1996--2000 and concurrent observations of the solar wind and interplanetary magnetic field (IMF). The essence of the approach is to derive from the data the temporal variation of all major current systems contributing to the distant geomagnetic field during the entire storm cycle, using a simple model of their growth and decay. Each principal source of the external magnetic field (magnetopause, cross-tail current sheet, axisymmetric and partial ring currents, and Birkeland current systems) is driven by a separate variable, calculated as a time integral of a combination of geoeffective parameters NλV¿Bsγ, where N, V, and Bs are the solar wind density, speed, and the magnitude of the southward component of the IMF, respectively. In this approach we assume that each source has its individual relaxation timescale and residual quiet-time strength, and its partial contribution to the total field depends on the entire history of the external driving of the magnetosphere during a storm. In addition, the magnitudes of the principal field sources were assumed to saturate during extremely large storms with abnormally strong external driving. All the parameters of the model field sources, including their magnitudes, geometrical characteristics, solar wind/IMF driving functions, decay timescales, and saturation thresholds, were treated as free variables, and their values were derived from the data. As an independent consistency test, we calculated the expected Dst variation on the basis of the model output at Earth's surface and compared it with the actual observed Dst. A good agreement (cumulative correlation coefficient R = 0.92) was found, in spite of the fact that ~90% of the spacecraft data used in the fitting were taken at synchronous orbit and beyond, while only 3.7% of those data came from distances 2.5 ≤ R ≤ 4 RE. The obtained results demonstrate the possibility to develop a truly dynamical model of the magnetic field, based on magnetospheric and interplanetary data and allowing one to reproduce and forecast the entire process of a geomagnetic storm, as it unfolds in time and space.