In this study, we proposed a machine learning-based system to distinguish patients with COVID-19 from non-COVID-19 patients by analyzing only a single cough sound. Two different data sets were used, one accessible for the public and the other available on request. After combining the data sets, the features were obtained from the cough sounds using the mel-frequency cepstral coefficients (MFCCs) method, and then they were classified with seven different machine learning classifiers. To determine the optimum values of hyperparameters for MFCCs and classifiers, the leave-one-out cross-validation (LOO-CV) strategy was implemented. Based on the results, the k-nearest neighbors classifier based on the Euclidean distance (k-NN Euclidean) with the accuracy rate, sensitivity of COVID-19, sensitivity of non-COVID-19, F-measure, and area under the ROC curve (AUC) of 0.9833, 1.0000, 0.9720, 0.9799, and 0.9860, respectively, is more successful than other classifiers. Finally, the best and most effective features were determined for each classifier using the sequential forward selection (SFS) method. According to the results, the proposed system is excellent compared with similar studies in the literature and can be easily used in smartphones and facilitate the diagnosis of COVID-19 patients. In addition, since the used data set includes reflex and unconscious coughs, the results showed that conscious or unconscious coughing has no effect on the diagnosis of COVID-19 patients based on the cough sound.