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Fully Homomorphic Encryption (FHE) allows for secure computation on encrypted data. Unfortunately, huge memory size, computational cost and bandwidth requirements limit its practicality. We present BASALISC, an architecture family of hardware accelerators that aims to substantially accelerate FHE computations in the cloud. BASALISC is the first to implement the BGV scheme with fully-packed bootstrapping -- the noise removal capability necessary for arbitrary-depth computation. It supports a customized version of bootstrapping that can be instantiated with hardware multipliers optimized for area and power. BASALISC is a three-abstraction-layer RISC architecture, designed for a 1 GHz ASIC implementation and underway toward 150mm2 die tape-out in a 12nm GF process. BASALISC's four-layer memory hierarchy includes a two-dimensional conflict-free inner memory layer that enables 32 Tb/s radix-256 NTT computations without pipeline stalls. Its conflict-resolution permutation hardware is generalized and re-used to compute BGV automorphisms without throughput penalty. BASALISC also has a custom multiply-accumulate unit to accelerate BGV key switching. The BASALISC toolchain comprises a custom compiler and a joint performance and correctness simulator. To evaluate BASALISC, we study its physical realizability, emulate and formally verify its core functional units, and we study its performance on a set of benchmarks. Simulation results show a speedup of more than 5,000 times over HElib -- a popular software FHE library.