We consider the continuous time symmetric random walk with a slow bond on $\mathbb Z$, which rates are equal to $1/2$ for all bonds, except for the bond of vertices $\{-1,0\}$, which associated rate is given by $αn^{-β}/2$, where $α\geq 0$ and $β\in [0,\infty]$ are the parameters of the model. We prove here a functional central limit theorem for the random walk with a slow bond: if $β<1$, then it converges to the usual Brownian motion. If $β\in (1,\infty]$, then it converges to the reflected Brownian motion. And at the critical value $β=1$, it converges to the snapping out Brownian motion (SNOB) of parameter $κ=2α$, which is a Brownian type-process recently constructed in Lejay, A., The snapping out Brownian motion. Ann. Appl. Probab., 26(3):1727--1742, 2016. We also provide Berry-Esseen estimates in the dual bounded Lipschitz metric for the weak convergence of one-dimensional distributions, which we believe to be sharp.