We consider a stochastic spatial point process with births and deaths on $\mathbb{R}^d$, with the hard-core property that at any time the balls of radius half of any two points do not overlap. We give explicit construction of the process. Under some more conditions, we show the stochastic stability of this Markov process by constructing a stationary regime for the dynamics. The main tool used to construct the stationary regime is the coupling form the past technique. Further, we study the exponential convergence of the probability distribution to the stationary distribution.