Changes and progresses in information technologies have played an important role in the development of intelligent vehicle systems in recent years. Driver fatigue is an important factor in vehicle accidents. For this reason, traffic accidents involving driver fatigue and driver carelessness have been followed by researchers. In this article, a Multi-tasking Convulational Neural Network (ConNN*) model is proposed to detect driver drowsiness/fatigue. Eye and mouth characteristics are utilized for driver's behavior model. Changes to these characteristics are used to monitor driver fatigue. With the proposed Multi-task ConNN model, unlike the studies in the literature, both mouth and eye information are classified into a single model at the same time. Driver fatigue is determined by calculating eyes' closure duration/Percentage of eye closure (PERCLOS) and yawning frequency/frequency of mouth (FOM). In this study, the fatigue degree of the driver is divided into 3 classes. The proposed model achieved 98.81% fatigue detection on YawdDD and NthuDDD dataset. The success of the model is presented comparatively.