Many of the most significant advances in accelerator science have been due to improvements in our ability to manipulate beam phase space. Despite steady progress in beam phase-space manipulation over the last several decades, future accelerator applications continue to outpace the ability to manipulate the phase space. This situation is especially pronounced for longitudinal beam phase-space manipulation, and is now getting increased attention. Herein, we report the first experimental demonstration of the double emittance exchange concept, which allows for the control of the longitudinal phase space using relatively simple transverse manipulation techniques. The double emittance exchange beamline enables extensive longitudinal manipulation, including tunable bunch compression, time-energy correlation control, and nonlinearity correction, in a remarkably flexible manner. The demonstration of this new method opens the door for arbitrary longitudinal beam manipulations capable of responding to the ever increasing demands of future accelerator applications.