A promising direction for enabling private queries to large language models (LLMs) is with homomorphic encryption (HE). An open problem is performing token sampling under HE. In this paper, we introduce Hyperion, an efficient HE algorithm for inverse transform sampling, enabling private token sampling with 1 comparison depth, O(1) amortized comparisons, and O(log n) rotations. We implement our approach and demonstrate that it samples tokens in 0.14 seconds for 32k tokens (≈ 4.4\ 𝜇 s per token) on GPU, achieving a 100× latency improvement over prior work.