Sub-Saharan Africa (SSA) needs additional affordable and reliable electricity to fuel its social and economic development. Ideally, all of this new supply is carbon-neutral. The potentials for renewables in SSA suffice for any conceivable demand, but the wind power and photovoltaic resources are intermittent and difficult to integrate in the weak electricity grids. Here, we investigate the potential for supplying SSA demand centers with dispatchable electricity from concentrating solar power (CSP) stations equipped with thermal storage. We show that, given anticipated cost reductions from technological improvements, power from CSP could be competitive with coal power in Southern Africa by 2025; but in most SSA countries, power from CSP may not be competitive. We also show that variations in risk across countries influences the cost of power from CSP more than variations in solar resources. If policies to de-risk CSP investment to financing cost levels found in industrialized countries were successfully implemented, power from CSP could become cheaper than coal power by 2025 in all SSA countries. Policies to increase institutional capacity and cooperation among SSA countries could reduce costs further. With dedicated policy measures, therefore, CSP could become an economically attractive electricity option for all SSA countries.