Constant Depth Circuit Computing Threshold Function
General weight threshold gates can be computed by polynomial size depth 2 circuits built from majority gates. An efficient construction of this is e.g. given by Amano and Maruoka. Then you can just compute each of these by constant depth circuits built from AND and OR gates.
You can always assume weights are integers of magnitude at most $(n+1)^{(n+1)/2}/2^n$. See Muroga, Toda and Takasu