Modern cosmology predicts that matter in our Universe has assembled today into a vast network of filamentary structures colloquially termed the Cosmic Web. Because this matter is either electromagnetically invisible (i.e., dark) or too diffuse to image in emission, tests of this cosmic web paradigm are limited. Wide-field surveys do reveal web-like structures in the galaxy distribution, but these luminous galaxies represent less than 10% of baryonic matter. Statistics of absorption by the intergalactic medium (IGM) via spectroscopy of distant quasars support the model yet have not conclusively tied the diffuse IGM to the web. Here, we report on a new method inspired by the Physarum polycephalum slime mold that is able to infer the density field of the Cosmic Web from galaxy surveys. Applying our technique to galaxy and absorption-line surveys of the local Universe, we demonstrate that the bulk of the IGM indeed resides in the Cosmic Web. From the outskirts of Cosmic Web filaments, at approximately the cosmic mean matter density (rho_m) and approx. 5 virial radii from nearby galaxies, we detect an increasing H I absorption signature towards higher densities and the circumgalactic medium, to approx. 200 rho_m. However, the absorption is suppressed within the densest environments, suggesting shock-heating and ionization deep within filaments and/or feedback processes within galaxies., Published in ApJL, 891, L35