We report the observation of intrinsic one-dimensional (ID) superconductivity in 0.4 nm-sized single-walled carbon nanotubes (SWNTs), fabricated by pyrolyzing tripropylamine in the channels of zeolite AIPO -5 single crystals. These ultrasmall SWNTs are highly aligned, uniform in size, and isolated from each other. They constitute an almost ideal 1D conducting system. The superconductivity of these ultra-small exhibits one-dimensional fluctuations, and displays smooth temperature variations with a mean-field superconducting transition temperature of 15 K. The data are consistent with the manifestations of a 1D BCS (phononmediated) superconductor.