There is welding cable that is dual listed as RHH and can be used for electrical systems, but you have to use the NEC ampacities. The manufacturers use the 90°C rating from Table 310.17. In most cases you are limited to the 75°C rating from Table 310.16. Also as Bob pointed out, you have to use a connector that is listed for use with the fine strands in the welding cable.
The listing of welding cable does bring up an issue to me. A few years ago, I was doing an emergency feed to an industrial building and we used welding cable as many do for this purpose. I did some reaseach on the welding cable and Type W power cable and found that the only difference was that the insulation on the Type W power cable was thinner than that on the welding cable. Every thing else from the stranding to the type of material used for the insulation was identical except the use of the welding cable with the better insulation was a code violation. Note that both the welding calbe and the Type W cable were made by the same company and I don't not know if this is true in general for all welding cable.
The code only applies to T&M work....it does not apply to contract work.