In this paper, a new three-parameter lifespan distribution based on the Gompertz distribution, called the Marshall-Elkin Gompertz, which is a generalization of the Gompertz distribution and has decreasing, increasing, and vane-shaped failure rates, is introduced, and the probability density function, cumulative distribution function, hazard function, and some properties of this new distribution, such as central moments, moments of order statistics, Rennie and Shannon entropies, and quantile function, are obtained. Also, its parameters are estimated using the maximum likelihood method, and with the help of a real dataset, this new distribution is compared with some lifespan distributions based on the Gompertz distribution.