Payment channel networks (PCNs) provide a faster and cheaper alternative to transactions recorded on the blockchain. Clients can trustlessly establish payment channels with relays by locking coins and then send signed payments that shift coin balances over the network’s channels. Although payments are never published, anyone can track a client’s payment by monitoring changes in coin balances over the network’s channels. We present Twilight, the first PCN that provides a rigorous differential privacy guarantee to its users. Relays in Twilight run a noisy payment processing mechanism that hides the payments they carry. This mechanism increases the relay’s cost, so Twilight combats selfish relays that wish to avoid it using a trusted execution environment (TEE) that ensures they follow its protocol. The TEE does not store the channel’s state, which minimizes the trusted computing base. Crucially, Twilight ensures that even if a relay breaks the TEE’s security, it cannot break the integrity of the PCN. We analyze Twilight in terms of privacy and cost and study the trade-off between them. We implement Twilight using Intel’s SGX framework and evaluate its performance using relays deployed on two continents. We show that a route consisting of 4 relays handles 820 payments/sec.