Spindle-shaped nematic droplets (tactoids) form in solutions of rod-like molecules at the onset of the liquid crystalline phase. Their unique shape and internal structure result from the interplay of the elastic deformation of the nematic and anisotropic surface forces. The balance of these forces dictates that tactoids must display a continuous variation in aspect ratio and director-field configuration. Yet, such continuous transition has eluded observation for decades: tactoids have displayed either a bipolar configuration with particles aligned parallel to the droplet interface or a homogeneous configuration with particles aligned parallel to the long axis of the tactoid. Here, we report the first observation of the continuous transition in shape and director-field configuration of tactoids in true solutions of carbon nanotubes in chlorosulfonic acid. This observation is possible because the exceptional length of carbon nanotubes shifts the transition to a size range that can be visualized by optical microscopy. Polarization micrographs yield the interfacial and elastic properties of the system. Absorbance anisotropy measurements provide the highest nematic order parameter (S=0.79) measured to date for a nematic phase of carbon nanotubes at coexistence with its isotropic phase.