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Atomic clocks can measure the gravitational redshift predicted by general relativity with great accuracy and for height differences as little as 1 cm. All existing experiments, however, involve the comparison of two independent clocks at different locations rather than a single clock in a delocalized quantum superposition. Here we present an interferometry scheme employing group-II-type atoms, such as Sr or Yb, capable of measuring the gravitational time dilation in a coherent superposition of atomic wave packets at two different heights. In contrast to other recent proposals, there is no need for pulses that can efficiently diffract both internal states. Instead, the scheme relies on very simple atom optics for which high diffraction efficiencies can be achieved with rather mild requirements on laser power. Furthermore, the effects of vibration noise are subtracted by employing a simultaneous Rb interferometer that acts as an inertial reference. Remarkably, the recently commissioned VLBAI facility in Hannover, a 10-m atomic fountain that can simultaneously operate Yb and Rb atoms and enables up to 2.8 s of free evolution time, meets all the requirements for a successful experimental implementation. [ABSTRACT FROM AUTHOR]