The task of video moment retrieval (VMR) is to retrieve the specific video moment from an untrimmed video, according to a textual query. It is a challenging task that requires effective modeling of complex cross-modal matching relationship. Recent efforts primarily model the cross-modal interactions by hand-crafted network architectures. Despite their effectiveness, they rely heavily on expert experience to select architectures and have numerous hyperparameters that need to be carefully tuned, which significantly limit their applications in real-world scenarios. How to design flexible architectures for modeling cross-modal interactions with less manual effort is crucial for the task of VMR but has received limited attention so far. To address this issue, we present a novel VMR approach that automatically searches for an optimal architecture to learn cross-modal matching relationship. Specifically, we develop a cross-modal architecture searching method. It first searches for repeatable cell network architectures based on a directed acyclic graph, which performs operation sampling over a customized task-specific operation set. Then, we adaptively modulate the edge importance in the graph by a query-aware attention network, which performs edge sampling softly in the searched cell. Different from existing neural architecture search methods, our approach can effectively exploit the query information to reach query-conditioned architectures for modeling cross modal matching. Extensive experiments on three benchmark datasets show that our approach can not only significantly outperform the state-of-the-art methods but also run more efficiently and robustly than manually crafted network architectures.