A sequence of low-energy levels in _{32}^{78}Ge_{46} has been identified with spins and parity of 2^{+}, 3^{+}, 4^{+}, 5^{+}, and 6^{+}. Decays within this band proceed strictly through ΔJ=1 transitions, unlike similar sequences in neighboring Ge and Se nuclei. Above the 2^{+} level, members of this sequence do not decay into the ground-state band. Moreover, the energy staggering of this sequence has the phase that would be expected for a γ-rigid structure. The energies and branching ratios of many of the levels are described well by shell-model calculations. However, the calculated reduced transition probabilities for the ΔJ=2 in-band transitions imply that they should have been observed, in contradiction with the experiment. Within the calculations of Davydov, Filippov, and Rostovsky for rigid-triaxial rotors with γ=30°, there are sequences of higher-spin levels connected by strong ΔJ=1 transitions which decay in the same manner as those observed experimentally, yet are calculated at too high an excitation energy.