What type of fuel pump are you using? The weber requires a low pressure pump 2-3PSI. The SU pump that came stock with your MGB works great, since it's output is 2.7PSI. If don't have an SU pump the fuel pressure may be too high, if so the needle valve and float in the weber will not close and the fuel bowl will completely fill.
Have the correct fuel pressure? Then I would check the float and needle valve to ensure it is to spec. Here's a great link on how to do that: http://www.piercemanifolds.com/Float_Level_1.htm