High contrast imaging is the primary path to the direct detection and characterization of Earth-like planets around solar-type stars; a cleverly designed internal coronagraph suppresses the light from the star, revealing the elusive circumstellar companions. However, future large-aperture telescopes ($>$4~m in diameter) will likely have segmented primary mirrors, which causes additional diffraction of unwanted stellar light. Here we present the first high contrast laboratory demonstration of an apodized vortex coronagraph (AVC), in which an apodizer is placed upstream of a vortex focal plane mask to improve its performance with a segmented aperture. The gray-scale apodization is numerically optimized to yield a better sensitivity to faint companions assuming an aperture shape similar to the LUVOIR-B concept. Using wavefront sensing and control over a one-sided dark hole, we achieve a raw contrast of $2times10^{-8}$ in monochromatic light at 775~nm, and a raw contrast of $4times10^{-8}$ in a 10% bandwidth. These results open the path to a new family of coronagraph designs, optimally suited for next-generation segmented space telescopes.