Encoding a qubit in logical quantum states with wave functions characterized by disjoint support and robust energies can offer simultaneous protection against relaxation and pure dephasing. One of the most promising candidates for such a fully protected superconducting qubit is the 0-π circuit [Brooks et al., Phys. Rev. A 87, 052306 (2013)]. Here we realize the proposed circuit topology in an experimentally obtainable parameter regime, where the ground-state degeneracy is lifted but the qubit is still largely noise protected. More precisely, the logical states of this qubit feature disjoint support and are exponentially protected against relaxation and exponentially (first order) protected against dephasing due to charge (flux) noise. We name the resultant device the "soft 0-π qubit."Multitone spectroscopy measurements reveal the energy-level structure of the system, which can be precisely described by a simple two-mode Hamiltonian. Using a Raman-type protocol, we exploit a higher-lying charge-insensitive energy level of the device to realize coherent population transfer and logical operations. The measured relaxation (T₁ = 1.6 ms) and dephasing (T_R = 9 μs, T_2E = 25μs) times demonstrate that the soft 0-π circuit not only broadens the family of superconducting qubits but also constitutes an important step toward quantum computing with intrinsically protected superconducting qubits.