Short Message Service (SMS) still continues its existence despite the emergence of different messaging services. It plays a part in our lives as a communication service. Companies use SMS for advertisement purposes due to the fact that e-mail filtering systems have rooted, short message systems are being undersold by the operators, and spam detection and blocking systems used for short messages are ineffective. Individuals falling victim to SMS spam messages sent by malevolent persons incur pecuniary and non-pecuniary losses. The aim of this study is to present a hybrid model proposal with the intention of detecting SMS spam messages. This detection model uses a gated recurrent unit (GRU) and convolutional neural network (CNN) as two deep learning methods. However, the fact that both algorithms require high memory capacities is a limitation. The design for this model was laid out by using two different datasets containing combined text messages written in the Turkish and English languages. The datasets used in the study are TurkishSMSCollection and the SMS Spam dataset from the UCI database. The testing process was performed on the dataset through benchmarking as well as other machine learning algorithms. It was revealed in the study that the hybrid CNN + GRU approach attained an accuracy of 99.07% by demonstrating a better performance compared to the other algorithms.