Large language models (LLMs) encode knowledge with varying degrees of confidence. When responding to queries, models face an inherent trade-off: they can generate responses that are less informative but highly factual, or more informative but potentially less accurate. Different applications demand different balances between informativeness and factuality. We introduce Factuality-Controlled Generation (FCG), a framework that enables users to specify factuality constraints alongside their queries. We propose to evaluate FCG performance on two dimensions: adherence to factuality constraints and response informativeness. We propose to train models on the FCG task using synthetic data, and show that our synthetic training significantly improves models' ability to both respect factuality requirements and maintain informativeness in their outputs.