Nanostructured carbons are posited to offer an alternative to silicon and lead to further miniaturization of photonic and electronic devices. Here, we report the experimental realization of the first all-carbon solid-state optical diode that is based on axially asymmetric nonlinear absorption in a thin saturable absorber (graphene) and a thin reverse saturable absorber (C60) arranged in tandem. This all-optical diode action is polarization independent and has no phase-matching constraints. The nonreciprocity factor of the device can be tuned by varying the number of graphene layers and the concentration or thickness of the C60 coating. This ultracompact graphene/C60 based optical diode is versatile with an inherently large bandwidth, chemical and thermal stability, and is poised for cost-effective large-scale integration with existing fabrication technologies.