Sulfuric acid curing and leaching is a promising technology for treating refractory ores. In this work, a refractory concentrate containing 3191 ppm uranium (U), 2135 ppm niobium (Nb), and 0.7% rare earth minerals (REMs) went through two stages: curing by high-concentration H2SO4 and leaching by low-concentration H2SO4. We investigated the behavior of those valuable metals during the two stages. For both curing and leaching, the operating parameters include the acid-to-solid ratio, time, temperature, and H2SO4 concentration. The recovery for U, Nb, and REMs was as high as 95%, 86%, and 73.5% using a curing acid-to-solid ratio of 1:1, curing temperature of 200 °C, curing time of 1 h, H2SO4 concentration of 98%, leaching liquid-to-solid ratio of 4:1, leaching time of 2 h, leaching temperature of 60 °C, and leaching H2SO4 concentration of 5 g/L. A “sulfuric acid curing–leaching-U extraction by N235–Nb recovery by resin adsorption–REMs’ recovery by resin adsorption” method was implemented, where the overall U, Nb, and REMs’ recovery reached 93.1%, 84.5%, and 69.6%, respectively.