MLE and MME for the exponential rate (again)

Let $x_1,x_2,\dots,x_n$ constitute a random sample of size $n$ from the probability density function $f(x;\theta)=\theta e^{-\theta x};\ 0<x<\infty, \theta>0$. Consider the following statements:

I. The estimator of $\theta$ with method of moments is $\bar x$.

II. Both maximum likelihood estimator and moment estimator of $\theta$ are same.

Which of the statements given above is/are correct?