In this paper we present a magnetic gradiometer based on a single core orthogonal fluxgate in fundamental mode. Contrarily to similar gradiometers based on detached sensor heads the use of a single core strongly reduces the offset instability of the output. The offset drift is caused by temperature variations and in a single ferromagnetic core the temperature is more uniform than in two detached cores. As a result the offset of both coils drift in a very similar way and when computing their difference to obtain the gradiometric signal the output is very stable. Moreover, we show that using a single core we have noise suppression due to correlation of the magnetic noise and this mechanism also leads to noise reduction of the gradiometric output. As a result we achieved 10 pT/m/VHz noise at 1 Hz.