Deploying large language models (LLMs) on edge platforms is challenged by their high computational and memory demands. Although recent low-bit quantization methods (e.g., BitNet, DeepSeek) compress weights to as little as 1.58 bits with minimal accuracy loss, edge deployment is still constrained by limited on-chip resources, power budgets, and the often-neglected latency of the prefill phase. We present TeLLMe, the first ternary LLM accelerator for low-power FPGAs (e.g., AMD KV260) that fully supports both prefill and autoregressive decoding using 1.58-bit weights and 8-bit activations. Our contributions include: (1) a table-lookup matrix engine for ternary matmul that merges grouped activations with online precomputation to minimize resource use; (2) a fused, bandwidth-efficient attention module featuring a reversed reordering scheme to accelerate prefill; and (3) a tightly integrated normalization and quantization--dequantization unit optimized for ultra-low-bit inference. Under a 7W power budget, TeLLMe delivers up to 9 tokens/s throughput over 1,024-token contexts and prefill latencies of 0.55--1.15 s for 64--128 token prompts, marking a significant energy-efficiency advance and establishing a new edge FPGA benchmark for generative AI.