An emerging strategy to remove CO2 from the atmosphere and compensate for the greenhouse-gas emissions of cement and concrete is based on incorporating biochar into concrete. With this approach, concrete can be turned into a functional carbon sink (C-sink). Until now, biochar has been mainly used without modification to replace part of the cement or of the aggregates in concrete. However, this technology comes with a number of practical problems, which include the high water absorption of the biochar (due to its high specific surface) and hazards (dust, risk of combustion). In this paper we present an alternative approach, in which biochar is first processed into lightweight aggregates in a cold-bonding process. To this end, biochar is pelletized together with water and a small amount of hydraulic binder forming round pellets that further harden with hydration. In this way, carbon-rich lightweight aggregates (C-LWA) are obtained that are easier to handle than the original biochar. The C-LWA pellets have similar porosity and strength as conventional LWA and can be used for similar applications. Yet, the CO2 emissions from sintering traditional LWA are avoided and the C-LWA are instead an effective C-sink. We demonstrate that it is possible to incorporate in the pellets and eventually in the concrete a sufficient amount of carbon to compensate for the original emissions of concrete. The net-zero emissions concrete obtained with this approach possesses mechanical performance sufficient for typical structural applications in buildings.