  We study the closure $\bar{CSO}$ of the set $CSO$ of all complex symmetric
operators on a separable, infinite-dimensional, complex Hilbert space. Among
other things, we prove that every compact operator in $\bar{CSO}$ is complex
symmetric. Using a construction of Kakutani as motivation, we also describe
many properties of weighted shifts in $\bar{CSO} \backslash CSO$. In
particular, we show that weighted shifts which demonstrate a type of
approximate self-similarity belong to $\bar{CSO}\backslash CSO$. As a byproduct
of our treatment of weighted shifts, we explain several ways in which our
result on compact operators is optimal.
