We present a compact atomic clock interrogating ultracold 87Rb magnetically trapped on an atom chip. Very long coherence times sustained by spin self-rephasing allow us to interrogate the atomic transition with 85% contrast at 5-s Ramsey time. The clock exhibits a fractional frequency stability of 5.8 ×10-13 at 1 s and is likely to integrate into the 10-15 range in less than a day. A detailed analysis of seven noise sources explains the measured frequency stability. Fluctuations in the atom temperature (0.4 nK shot-to-shot) and in the offset magnetic field (5 ×10-6 relative fluctuations shot-to-shot) are the main noise sources together with the local oscillator, which is degraded by the 30% duty cycle. The analysis suggests technical improvements to be implemented in a future second generation setup . The results demonstrate the remarkable degree of technical control that can be reached in an atom chip experiment.