Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) has caused outbreaks of new coronavirus disease (COVID-19) around the world. Rapid and accurate detection of COVID-19 coronavirus is an important step in limiting the spread of the COVID-19 epidemic. To solve this problem, radiography techniques (such as chest X-rays and computed tomography (CT)) can play an important role in the early prediction of COVID-19 patients, which will help to treat patients in a timely manner. We aimed to quickly develop a highly efficient lightweight CNN architecture for detecting COVID-19-infected patients. The purpose of this paper is to propose a robust deep learning-based system for reliably detecting COVID-19 from chest X-ray images. First, we evaluate the performance of various pre-trained deep learning models (InceptionV3, Xception, MobileNetV2, NasNet and DenseNet201) recently proposed for medical image classification. Second, a lightweight shallow convolutional neural network (CNN) architecture is proposed for classifying X-ray images of a patient with a low false-negative rate. The data set used in this work contains 2,541 chest X-rays from two different public databases, which have confirmed COVID-19 positive and healthy cases. The performance of the proposed model is compared with the performance of pre-trained deep learning models. The results show that the proposed shallow CNN provides a maximum accuracy of 99.68% and more importantly sensitivity, specificity and AUC of 99.66%, 99.70% and 99.98%. The proposed model has fewer parameters and low complexity compared to other deep learning models. The experimental results of our proposed method show that it is superior to the existing state-of-the-art methods. We believe that this model can help healthcare professionals to treat COVID-19 patients through improved and faster patient screening.