We present a new first-order autoregressive time series model (so-called NUAR(1) model) for continuous uniform $(0,1)$ variables, given by $ X_n=\begin{cases} \alpha X_{n-1}, & \text{ w.p. } \alpha,\\ \beta X_{n-1}+\varepsilon_n, & \text{ w.p. } 1-\alpha, \end{cases} $ where $0<\alpha,\beta<1$, $(1-\alpha)/\beta\in\{1,2,\dots\}$ and $\{\varepsilon_n\}$ is the innovation sequence of independent and identically distributed random variables, such that each $X_n$ has continuous uniform $(0,1)$ distribution. The distribution of the innovation sequence and autoregressive structure of NUAR(1) model are discussed. It is shown that this model is partially time-reversible if the parameters are equal. We give also the estimates of the parameters of the model.