Biological fixation of atmospheric CO2 via the Calvin-Benson-Bassham cycle has massive ecological impact and offers potential for industrial exploitation, either by improving carbon fixation in plants and autotrophic bacteria, or by installation into new hosts. A kinetic model of the Calvin-Benson-Bassham cycle embedded in the central carbon metabolism of the cyanobacterium Synechocystis sp. PCC 6803 was developed to investigate its stability and underlying control mechanisms. To reduce the uncertainty associated with a single parameter set, random sampling of the steady-state metabolite concentrations and the enzyme kinetic parameters was employed, resulting in millions of parameterized models which were analyzed for flux control and stability against perturbation. Our results show that the Calvin cycle had an overall high intrinsic stability, but a high concentration of ribulose 1,5-bisphosphate was associated with unstable states. Low substrate saturation and high product saturation of enzymes involved in highly interconnected reactions correlated with increased network stability. Flux control, that is the effect that a change in one reaction rate has on the other reactions in the network, was distributed and mostly exerted by energy supply (ATP), but also by cofactor supply (NADPH). Sedoheptulose 1,7-bisphosphatase/fructose 1,6-bisphosphatase, fructose-bisphosphate aldolase, and transketolase had a weak but positive effect on overall network flux, in agreement with published observations. The identified flux control and relationships between metabolite concentrations and system stability can guide metabolic engineering. The kinetic model structure and parameterizing framework can be expanded for analysis of metabolic systems beyond the Calvin cycle.