Join Lists in Turns (Weave ?)

How can I put list A (0,1,2,3) and list B (0,1,2,3) together in turns to a list (0,1,2,3,4,5,6,7,8) ?
I tried weave flatten and sub list so far but it seems like there is a really easy solution to this ?
Appreciate the help

If not this, what do you expect to see?