A process for removing labile and refractory sulfur from a sulfur-bearing hydrocarbon stream derived from one or more naturally-occurring hydrocarbon sources, which process comprises (i) reacting said stream with hydrogen, in the presence of a hydrodesulfurizing catalyst and hydrogen, at reaction conditions effective to remove most of the labile sulfur, but not the refractory sulfur, and produce a sulfur-reduced stream and H2S, (ii) separating said H2S from said sulfur-reduced stream to produce a stream reduced in H2S, and (iii) contacting said H2S reduced stream with a solid reagent comprising supported sodium having a free sodium content, to remove at least a portion of said refractory sulfur, and produce a second sulfur-reduced stream further reduced in sulfur.