I have this true false question on prior exams I am preparing for, is an ergodic process weakly stationary?

As far as I understand it, an ergodic process means that any two "Zt's" or variables are asymptotically independent so they are a bit similar but not much. A weakly stationary process means that the covariance between the two Zt's depens only on the distance in the time series, so the further away they are in time the more their covariance tends towards zero.

It seems the concepts are similar, the variance between the variables asymptotically fading away in the time series.

But can we say "An ergodic process IS weakly stationary" and why?