How much is the volume of urban bicycle traffic affected by the provision of bicycle infrastructure? We exploit a large dataset of observed bicycle trajectories in combination with a fine-grained representation of the Copenhagen bicycle-relevant network. We apply a novel model for the bicyclist choice of route from origin to destination that takes the complete network into account. This enables us to back out the bicyclist preferences for a range of infrastructure and land-use types. We use the estimated preferences to compute a subjective 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 increased the number of bicycle trips by 40 percent and the volume of bicycle km by 60 percent, compared to a counterfactual without the bicycle lane network. This translates into an annual benefit worth 0.4 M EUR per km of bicycle lane due to changes in subjective travel cost, health, and accidents. Our results thus strongly support the provision of bicycle infrastructure.