Homomorphic Encryption (HE) is one of the most promising post-quantum cryptographic schemes that enable privacy-preserving computation on servers. However, noise accumulates as we perform operations on HE-encrypted data, restricting the number of possible operations. Fully HE (FHE) removes this restriction by introducing the bootstrapping operation, which refreshes the data; however, FHE schemes are highly memory-bound. Bootstrapping, in particular, requires loading GBs of evaluation keys and plaintexts from off-chip memory, which makes FHE acceleration fundamentally bottlenecked by the off-chip memory bandwidth. In this paper, we propose ARK, an Accelerator for FHE with Runtime data generation and inter-operation Key reuse. ARK enables practical FHE workloads with a novel algorithm-architecture co-design to accelerate bootstrapping. We first eliminate the off-chip memory bandwidth bottleneck through runtime data generation and inter-operation key reuse. This approach enables ARK to fully exploit on-chip memory by substantially reducing the size of the working set. On top of such algorithmic enhancements, we build ARK microarchitecture that minimizes on-chip data movement through an efficient, alternating data distribution policy based on the data access patterns and a streamlined dataflow organization of the tailored functional units -- including base conversion, number-theoretic transform, and automorphism units. Overall, our co-design effectively handles the heavy computation and data movement overheads of FHE, drastically reducing the cost of HE operations, including bootstrapping.