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We demonstrate a proof-of-principle of direct Earth gravity-gradient measurement with an atom interferometer-based gravity gradiometer using a single proof mass of cold 87 Rb atoms. The atomic gradiometer is implemented in the so-called double-loop configuration, hence providing a direct gravity-gradient dependent phase shift insensitive to DC acceleration and constant rotation rate. The atom interferometer (AI) can be either operated as a gravimeter or a gradiometer by simply adding an extra Raman π-pulse. We demonstrate gravity-gradient measurements first using a vibration isolation platform and second without seismic isolation using the correlation between the AI signal and the vibration signal measured by an auxilliary classical accelerometer allowing to bypass the absence of common-mode vibration noise rejection in a double-loop geometry.The simplicity of the experimental setup (a single atomic source and unique detection) and the immunity of the AI to rotation-induced contrast loss, make it a possible candidate for onboard gravity-gradient measurements.