Though Dialogue State Tracking (DST) is a core component of spoken dialogue systems, recent work on this task mostly deals with chat corpora, disregarding the discrepancies between spoken and written language. In this paper, we propose OLISIA, a cascade system which integrates an Automatic Speech Recognition (ASR) model and a DST model. We introduce several adaptations in the ASR and DST modules to improve integration and robustness to spoken conversations. With these adaptations, our system ranked first in DSTC11 Track 3, a benchmark to evaluate spoken DST. We conduct an in-depth analysis of the results and find that normalizing the ASR outputs and adapting the DST inputs through data augmentation, along with increasing the pre-trained models size all play an important role in reducing the performance discrepancy between written and spoken conversations.