MLE and MME for the exponential rate

If $X\sim\operatorname{Exp}(\lambda)$, then based on a random sample of size $n$ from this population, consider the following statements:

I. MLE and MME estimators of $\lambda$ are equal to $\hat\lambda=\dfrac1{\bar x}$.

II. MLE and MME estimators of $\lambda$ are equal to $\hat\lambda=\bar x$.

III. MLE and MME of $\lambda$ are unequal.

Which of the statements given above is correct?