We give a linear-time algorithm that approximately uniformly generates a random simple graph with a power-law degree sequence whose exponent is at least 2.8811. While sampling graphs with power-law degree sequence of exponent at least 3 is fairly easy, and many samplers work efficiently in this case, the problem becomes dramatically more difficult when the exponent drops below 3; ours is the first provably practicable sampler for this case. We also show that with an appropriate rejection scheme, our algorithm can be tuned into an exact uniform sampler. The running time of the exact sampler is O(n^{2.107}) with high probability, and O(n^{4.081}) in expectation.