Atmospheric aerosols exert an important influence on climate through their effects on stratiform cloud albedo and lifetime and the invigoration of convective storms. Model calculations suggest that almost half of the global cloud condensation nuclei in the atmospheric boundary layer may originate from the nucleation of aerosols from trace condensable vapours, although the sensitivity of the number of cloud condensation nuclei to changes of nucleation rate may be small. Despite extensive research, fundamental questions remain about the nucleation rate of sulphuric acid particles and the mechanisms responsible, including the roles of galactic cosmic rays and other chemical species such as ammonia. Here we present the first results from the CLOUD experiment at CERN. We find that atmospherically relevant ammonia mixing ratios of 100 parts per trillion by volume, or less, increase the nucleation rate of sulphuric acid particles more than 100-1,000-fold. Time-resolved molecular measurements reveal that nucleation proceeds by a base-stabilization mechanism involving the stepwise accretion of ammonia molecules. Ions increase the nucleation rate by an additional factor of between two and more than ten at ground-level galactic-cosmic-ray intensities, provided that the nucleation rate lies below the limiting ion-pair production rate. We find that ion-induced binary nucleation of H(2)SO(4)-H(2)O can occur in the mid-troposphere but is negligible in the boundary layer. However, even with the large enhancements in rate due to ammonia and ions, atmospheric concentrations of ammonia and sulphuric acid are insufficient to account for observed boundary-layer nucleation.