We present a machine learning based COVID-19 cough classifier which can discriminate COVID-19 positive coughs from both COVID-19 negative and healthy coughs recorded on a smartphone. This type of screening is non-contact, easy to apply, and can reduce the workload in testing centres as well as limit transmission by recommending early self-isolation to those who have a cough suggestive of COVID-19. The datasets used in this study include subjects from all six continents and contain both forced and natural coughs, indicating that the approach is widely applicable. The publicly available Coswara dataset contains 92 COVID-19 positive and 1079 healthy subjects, while the second smaller dataset was collected mostly in South Africa and contains 18 COVID-19 positive and 26 COVID-19 negative subjects who have undergone a SARS-CoV laboratory test. Both datasets indicate that COVID-19 positive coughs are 15\%-20\% shorter than non-COVID coughs. Dataset skew was addressed by applying the synthetic minority oversampling technique (SMOTE). A leave-$p$-out cross-validation scheme was used to train and evaluate seven machine learning classifiers: LR, KNN, SVM, MLP, CNN, LSTM and Resnet50. Our results show that although all classifiers were able to identify COVID-19 coughs, the best performance was exhibited by the Resnet50 classifier, which was best able to discriminate between the COVID-19 positive and the healthy coughs with an area under the ROC curve (AUC) of 0.98. An LSTM classifier was best able to discriminate between the COVID-19 positive and COVID-19 negative coughs, with an AUC of 0.94 after selecting the best 13 features from a sequential forward selection (SFS). Since this type of cough audio classification is cost-effective and easy to deploy, it is potentially a useful and viable means of non-contact COVID-19 screening.