Weakly-supervised object detection (WSOD) models attempt to leverage image-level annotations in lieu of accurate but costly-to-obtain object localization labels. This oftentimes leads to substandard object detection and localization at inference time. To tackle this issue, we propose D2DF2WOD, a Dual-Domain Fully-to-Weakly Supervised Object Detection framework that leverages synthetic data, annotated with precise object localization, to supplement a natural image target domain, where only image-level labels are available. In its warm-up domain adaptation stage, the model learns a fully-supervised object detector (FSOD) to improve the precision of the object proposals in the target domain, and at the same time learns target-domain-specific and detection-aware proposal features. In its main WSOD stage, a WSOD model is specifically tuned to the target domain. The feature extractor and the object proposal generator of the WSOD model are built upon the fine-tuned FSOD model. We test D2DF2WOD on five dual-domain image benchmarks. The results show that our method results in consistently improved object detection and localization compared with state-of-the-art methods.