Distributing elements across a list of lists

I have two lists:

u = {{1, 3}, {2, 6}, {3, 9}}
v = {0, 4}

and I want to obtain this list from them:

z = {{{0, 1}, {4, 3}}, {{0, 2}, {4, 6}}, {{0, 3},{4, 9}}}

I guess the solution will make use of Map, Thread, or even MapThread but I've tried every combinaison I can think of with no luck. How can I do it?

Answer

This works:

Transpose[{v, #}] & /@ u

{{{0, 1}, {4, 3}}, {{0, 2}, {4, 6}}, {{0, 3}, {4, 9}}}