---

In this paper, a new integer-valued autoregressive process is introduced based on the discrete exponential-Weibull distribution to model integer-value time series data. Regarding the importance of discrete distributions in counting data modeling, the discrete counterpart of the exponential-Weibull distribution is introduced, and some of its statistical properties, such as survival function, hazard rate, moment generating function, skewness and kurtosis, are investigated. The Fisher dispersion, skewness and kurtosis indices show the flexibility and efficiency of the discrete Exponential-Weibull distribution in fitting different types of counting data. The discrete Exponential-Weibull distribution covers data fits with different dispersion characteristics (overdispersion, underdispersion and equidispersion), long right tail  (skewed to the right) and heavy-tailed. The model parameters are estimated using three approaches maximum conditional likelihood, minimum generalized conditional squares, and Yule-Walker. Finally, the efficiency and superiority of the process in fitting counts data of deaths due to COVID-19 disease are compared with other competing models.