The current clinical diagnosis of COVID-19 requires person-to-person contact, needs variable time to produce results, and is expensive. It is even inaccessible to the general population in some developing countries due to insufficient healthcare facilities. Hence, a low-cost, quick, and easily accessible solution for COVID-19 diagnosis is vital. This paper presents a study that involves developing an algorithm for automated and noninvasive diagnosis of COVID-19 using cough sound samples and a deep neural network. The cough sounds provide essential information about the behavior of glottis under different respiratory pathological conditions. Hence, the characteristics of cough sounds can identify respiratory diseases like COVID-19. The proposed algorithm consists of three main steps (a) extraction of acoustic features from the cough sound samples, (b) formation of a feature vector, and (c) classification of the cough sound samples using a deep neural network. The output from the proposed system provides a COVID-19 likelihood diagnosis. In this work, we consider three acoustic feature vectors, namely (a) time-domain, (b) frequency-domain, and (c) mixed-domain (i.e., a combination of features in both time-domain and frequency-domain). The performance of the proposed algorithm is evaluated using cough sound samples collected from healthy and COVID-19 patients. The results show that the proposed algorithm automatically detects COVID-19 cough sound samples with an overall accuracy of 89.2%, 97.5%, and 93.8% using time-domain, frequency-domain, and mixed-domain feature vectors, respectively. The proposed algorithm, coupled with its high accuracy, demonstrates that it can be used for quick identification or early screening of COVID-19. We also compare our results with that of some state-of-the-art works.