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 €0.4M 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.