Blind face restoration (BFR) may correspond to multiple plausible high-quality (HQ) reconstructions under extremely low-quality (LQ) inputs. However, existing methods typically produce deterministic results, struggling to capture this one-to-many nature. In this paper, we propose a Measurement-Constrained Sampling (MCS) approach that enables diverse LQ face reconstructions conditioned on different textual prompts. Specifically, we formulate BFR as a measurement-constrained generative task by constructing an inverse problem through controlled degradations of coarse restorations, which allows posterior-guided sampling within text-to-image diffusion. Measurement constraints include both Forward Measurement, which ensures results align with input structures, and Reverse Measurement, which produces projection spaces, ensuring that the solution can align with various prompts. Experiments show that our MCS can generate prompt-aligned results and outperforms existing BFR methods. Codes will be released after acceptance.