To what extent is the volume of urban bicycle traffic affected by the provision of bicycle infrastructure? In this study, we exploit a large dataset of GPS trajectories of bicycle trips in combination with a fine-grained representation of the Copenhagen bicycle-relevant network. We apply a model for bicyclists' choice of route from origin to destination that takes the complete network into account. This enables us to determine bicyclists' preferences for a range of infrastructure and land-use types. We use the estimated preferences to compute a generalized cost of bicycle travel, which we correlate with the number of bicycle trips across a large number of origin-destination pairs. Simulations suggest that the extensive Copenhagen bicycle lane network has caused the number of bicycle trips and the bicycle kilometers traveled to increase by 60% and 90%, respectively, compared with a counterfactual without the bicycle lane network. This translates into an annual benefit of e0.4M per km of bicycle lane owing to changes in generalized travel cost, health, and accidents. Our results thus strongly support the provision of bicycle infrastructure.