In this paper we present a weak limit theorem for a numerical approximation of Brownian semi-stationary processes studied in . In the original work of  the authors propose to use Fourier transformation to embed a given one dimensional (Lévy) Brownian semi-stationary process into a two-parameter stochastic field. For the latter they use a simple iteration procedure and study the strong approximation error of the resulting numerical scheme given that the volatility process is fully observed. In this work we present the corresponding weak limit theorem for the setting, where the volatility/drift process needs to be numerically simulated. In particular, weak approximation errors for smooth test functions can be obtained from our asymptotic theory.