There are two quick and dirty solutions. First would be to disaggregate series B to weekly values (R package **[tempdisagg][1]** is great for that) and then do a usual model. Second aggregate series A to monthly frequency, do a forecast and then use disaggregation on the forecast. 

The more theoretical approach would be casting problem to a state space model. There are a lot of literature on state space model approach when the dependent variable is observed at lower frequency. It usually assumes that the low frequency variable is really a high frequency variable observed at low frequency periods. You can make the same assumption and then reverse the methodology. Unfortunately I have not seen something similar being done, but I did not look hard enough.


  [1]: http://www.google.lt/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&ved=0CC4QFjAA&url=http://cran.r-project.org/package=tempdisagg&ei=SMP0UqCMOcnjoATy0oLICQ&usg=AFQjCNG3lrIsNP8nOwDwjiYXttMvvlT_Ow&sig2=lfRXfG44ct9yKu_RLfIY6Q&bvm=bv.60799247,d.cGU