Accurately understanding the interior structure of extra-solar planets is critical for inferring their formation and evolution and resolving the origin of anomalous planetary radii. The internal density distribution of the planet has a direct effect on the star-planet orbit through the gravitational quadrupole of rotational and tidal bulges, measured by the planetary Love number (k2p, twice the apsidal motion constant). We find that the quadrupole of the planetary tidal bulges dominates the rate of apsidal precession of single very hot Jupiters by more than an order of magnitude over general relativity and the stellar quadrupole. For the shortest-period planets, the planetary interior induces precession of a few degrees per year. By investigating the full photometric signal of apsidal precession, we find that transit timing induces a relatively small signal compared to the changes in transit shapes. With its long baseline of ultra-precise photometry, the future space-based Kepler mission should be able to realistically detect the presence or absence of a core in very hot Jupiters with orbital eccentricities as low as e ~ 0.001. We show that the signal due to k2p is not degenerate with other parameters and has a unique signature on the transit light curve. This technique, outlined in more detail in Ragozzine & Wolf 2008 provides the first readily employed method for directly probing the interiors of extra-solar planets.