Mass cannot be created or destroyed. Basic rule of non-nuclear physics (negating nuclear reaction type e=mc2 effects, which I'm sure is beyond the average Norweigan Blue).
Lift on birds' wings has to equal their weight, or they'd accelerate in a vertical direction (which would make it a slightly different case - cf jumping in lift, where f=ma also comes into the equation), though if they were all flapping up and down, the accelerations would average out.
Mass constant, acceleration constant, w = mg, therefore weight constant.