About half of present-day cloud condensation nuclei originate from atmospheric nucleation, frequently appearing as a burst of new particles near midday. Atmospheric observations show that the growth rate of new particles often accelerates when the diameter of the particles is between one and ten nanometres. In this critical size range, new particles are most likely to be lost by coagulation with pre-existing particles, thereby failing to form new cloud condensation nuclei that are typically 50 to 100 nanometres across. Sulfuric acid vapour is often involved in nucleation but is too scarce to explain most subsequent growth, leaving organic vapours as the most plausible alternative, at least in the planetary boundary layer. Although recent studies predict that low-volatility organic vapours contribute during initial growth, direct evidence has been lacking. The accelerating growth may result from increased photolytic production of condensable organic species in the afternoon, and the presence of a possible Kelvin (curvature) effect, which inhibits organic vapour condensation on the smallest particles (the nano-Köhler theory), has so far remained ambiguous. Here we present experiments performed in a large chamber under atmospheric conditions that investigate the role of organic vapours in the initial growth of nucleated organic particles in the absence of inorganic acids and bases such as sulfuric acid or ammonia and amines, respectively. Using data from the same set of experiments, it has been shown that organic vapours alone can drive nucleation. We focus on the growth of nucleated particles and find that the organic vapours that drive initial growth have extremely low volatilities (saturation concentration less than 10(-4.5) micrograms per cubic metre). As the particles increase in size and the Kelvin barrier falls, subsequent growth is primarily due to more abundant organic vapours of slightly higher volatility (saturation concentrations of 10(-4.5) to 10(-0.5) micrograms per cubic metre). We present a particle growth model that quantitatively reproduces our measurements. Furthermore, we implement a parameterization of the first steps of growth in a global aerosol model and find that concentrations of atmospheric cloud concentration nuclei can change substantially in response, that is, by up to 50 per cent in comparison with previously assumed growth rate parameterizations.