We present RICO, a code designed to compute the ionization fraction of the universe during the epoch of hydrogen and helium recombination with an unprecedented combination of speed and accuracy. This is accomplished by training the machine learning code PICO on the calculations of a multilevel cosmological recombination code which self-consistently includes several physical processes that were neglected previously. After training, RICO is used to fit the free electron fraction as a function of the cosmological parameters. While, for example, at low redshifts (z lsim 900), much of the net change in the ionization fraction can be captured by lowering the hydrogen fudge factor in RECFAST by about 3%, RICO provides a means of effectively using the accurate ionization history of the full recombination code in the standard cosmological parameter estimation framework without the need to add new or refined fudge factors or functions to a simple recombination model. Within the new approach presented here, it is easy to update RICO whenever a more accurate full recombination code becomes available. Once trained, RICO computes the cosmological ionization history with negligible fitting error in ~10 ms, a speedup of at least 106 over the full recombination code that was used here. Also RICO is able to reproduce the ionization history of the full code to a level well below 0.1%, thereby ensuring that the theoretical power spectra of cosmic microwave background (CMB) fluctuations can be computed to sufficient accuracy and speed for analysis from upcoming CMB experiments like Planck. Furthermore, it will enable cross-checking different recombination codes across cosmological parameter space, a comparison that will be very important in order to assure the accurate interpretation of future CMB data.