Generative models lack rigorous statistical guarantees with respect to their predictions. In this work, we propose Sequential Conformal Prediction for Generative Models (SCOPE-Gen), a sequential conformal prediction method producing prediction sets that satisfy a rigorous statistical guarantee called conformal admissibility control. This guarantee means that the prediction sets contain at least one admissible (or valid) example, with high probability. To this end, our method first samples an initial set of i.i.d. examples from a black box generative model. Then, this set is iteratively pruned via so-called greedy filters. As a consequence of the iterative generation procedure, admissibility of the final prediction set factorizes as a Markov chain, where each factor can be controlled separately, using conformal prediction. In comparison to prior work, our method demonstrates a large reduction in the number of admissibility evaluations during calibration. This is crucial e.g. in safety-critical applications, where these evaluations must be conducted manually by domain experts and are therefore costly and time consuming. We highlight the advantages of our method in terms of admissibility evaluations and cardinality of the prediction set through experiments in natural language generation and molecular graph extension tasks.