Let $\bar X$ denote the completion.
The weak topology $\sigma(X,X)$ on $X$ is strictly coarser than the weak topology $\sigma(X,\bar X)$; see [Schaefer, Topological Vector Spaces, (IV.1.2)].

However, both topologies have the same bounded subsets (which are the norm-bounded sets). 
And each equicontinuous subset of $X$ for the $\sigma(X,X)$-topology (i.e., norm bounded by the uniform boundedness theorem) the topologies $\sigma(X,X)$ and $\sigma(X,\bar X)$ coincide. And $\sigma(X,\bar X)$ is the finest locally convex topology on $X$ which coincides with $\sigma(X,X)$ on each norm-ball.