We consider the weakly asymmetric simple exclusion process in the presence of a slow bond and starting from the invariant state, namely the Bernoulli product measure of parameter $ρ\in(0,1)$. The rate of passage of particles to the right (resp. left) is $\frac1{\vphantom{n^β}2}+\frac{a}{2n^{\vphantomβγ}}$ (resp. $\frac1{\vphantom{n^β}2}-\frac{a}{2n^{\vphantomβγ}}$) except at the bond of vertices $\{-1,0\}$ where the rate to the right (resp. left) is given by $\fracα{2n^β}+\frac{a}{2n^{\vphantomβγ}}$ (resp. $\fracα{2n^β}-\frac{a}{2n^{\vphantomβγ}}$). Above, $α>0$, $γ\geq β\geq 0$, $a\geq 0$. For $β<1$, we show that the limit density fluctuation field is an Ornstein-Uhlenbeck process defined on the Schwartz space if $γ>\frac12$, while for $γ= \frac12$ it is an energy solution of the stochastic Burgers equation. For $γ\geqβ=1$, it is an Ornstein-Uhlenbeck process associated to the heat equation with Robin's boundary conditions. For $γ\geqβ> 1$, the limit density fluctuation field is an Ornstein-Uhlenbeck process associated to the heat equation with Neumann's boundary conditions.