Optical imaging plays a central role in biology and medicine but is hindered by light scattering in live tissue. We report the counterintuitive observation that strongly absorbing molecules can achieve optical transparency in live animals. We explored the physics behind this observation and found that when strongly absorbing molecules dissolve in water, they can modify the refractive index of the aqueous medium through the Kramers-Kronig relations to match that of high-index tissue components such as lipids. We have demonstrated that our straightforward approach can reversibly render a live mouse body transparent to allow visualization of a wide range of deep-seated structures and activities. This work suggests that the search for high-performance optical clearing agents should focus on strongly absorbing molecules.