Yes, but I don't see the point of putting down a new membrane. The weeds will come back again. Your better off with just the bark.

you could spray the weeds with a suitable weedkiller first [glyphosate] and then put membrane on top of the old one. then put the bark on top. I do find a membrane does work for most weeds but seed that lands between the bark will germinate no matter what. you need to keep hoeing the bark/mulch etc to remove seedlings as the germinate.

Yes agree with above, remember bark naturally breaks down so a seed bed is formed, may look pretty when you put all that bark down to start with but with neglect you have defeated the object, on really large areas then bark has its uses but small areas are best kept turned over and hoed, done so each week will keep the area looking cared for and cheaper long term than keeping it topped up with bark.