Modern exhaust-gas purification systems in motor vehicles with a lean-burn engine include a starting catalyst fitted close to the engine and a main catalyst arranged in the underbody region, with both the starting catalyst and the main catalyst being formed by nitrogen oxide storage catalysts. The nitrogen oxide storage catalysts are in each case regenerated by the engine being briefly switched from lean-burn mode to rich-burn mode when the nitrogen oxide concentration in the exhaust gas downstream of the storage catalysts rises above a predetermined value.The starting catalyst is exposed to particularly high temperatures and is therefore prone to faster ageing of its nitrogen oxide storage capacity than the main catalyst. To check the nitrogen oxide storage capacity of the starting catalyst, a regeneration which is due for the catalyst system is selected in such a way in terms of its duration and the extent to which the exhaust gas is enriched that substantially only the starting catalyst is regenerated, whereas the main catalyst is not. In this arrangement, the criterion for terminating this partial regeneration is the breakthrough of rich exhaust gas through the starting catalyst. After the engine has been switched back to lean-burn mode, the time which elapses until the concentration of nitrogen oxides in the exhaust gas downstream of the catalyst requires regeneration again is measured. The measured time is a measure of the remaining nitrogen oxide storage capacity of the starting catalyst.