We consider the component structure of a recent model of random graphs on the hyperbolic plane that was introduced by Krioukov et al. The model exhibits a power law degree sequence, small distances and clustering, features that are associated with the so-called complex networks. The model is controlled by two parameters $α$ and $ν$ where, roughly speaking, $α$ controls the exponent of the power law and $ν$ controls the average degree. Refining earlier results, we are able to show a law of large numbers for the largest component. That is, we show that the fraction of points in the largest component tends in probability to a constant $c$ that depends only on $α,ν$, while all other components are sublinear. We also study how $c$ depends on $α, ν$. To deduce our results, we introduce a local approximation of the random graph by a continuum percolation model on $\mathbb{R}^2$ that may be of independent interest.