MLE as a function of a sufficient statistic; invariance

Consider the following statements:

I. If a sufficient statistic $t$ for $\theta$ exists, then the maximum likelihood estimator will be a function of the sufficient statistic $t$.

II. If $\hat\theta$ is the maximum likelihood estimator of $\theta$, then $\dfrac{\hat\theta}{1+\hat\theta}$ is the maximum likelihood estimator of $\dfrac\theta{1-\theta}$.

Which of the statements given above is/are correct?