We present an analytical determination of the star formation rate (SFR) in molecular clouds, based on a time-dependent extension of our analytical theory of the stellar initial mass function. The theory yields SFRs in good agreement with observations, suggesting that turbulence is the dominant, initial process responsible for star formation. In contrast to previous SFR theories, the present one does not invoke an ad hoc density threshold for star formation; instead, the SFR continuously increases with gas density, naturally yielding two different characteristic regimes, thus two different slopes in the SFR versus gas density relationship, in agreement with observational determinations. Besides the complete SFR derivation, we also provide a simplified expression, which reproduces the complete calculations reasonably well and can easily be used for quick determinations of SFRs in cloud environments. A key property at the heart of both our complete and simplified theory is that the SFR involves a density-dependent dynamical time, characteristic of each collapsing (prestellar) overdense region in the cloud, instead of one single mean or critical freefall timescale. Unfortunately, the SFR also depends on some ill-determined parameters, such as the core-to-star mass conversion efficiency and the crossing timescale. Although we provide estimates for these parameters, their uncertainty hampers a precise quantitative determination of the SFR, within less than a factor of a few.