Given a right-censored (RC) data set, it is desirable to make parametric inferences. One class of the distribution families is the uniform distribution $U(a,b)$ ($0\le a< b< \infty$). We derive the MLE with RC data under this parametric assumption. Under the survival context, it is rare that one can find the explicit solution to the MLE under the parametric set-up. One exception is the exponential distribution. It turns out that the MLE of $a$ has a closed form solution if $a$ is a parameter and the MLE of $b$ has a closed form solution in some sense if $b$ is a parameter. It is shown that the MLE is consistent under the necessary and sufficient identifiability condition.