  In this paper we prove the following: let $\omega(t)$ be a continuous
function, increasing in $[0,\infty)$ and $\omega(+0)=0$. Then there exists a
series of the form$\sum_{k=-\infty}^\infty C_ke^{ikx}$ with
$\sum_{k=-\infty}^\infty C^2_k \omega(|C_k|)<\infty$, $C_{-k}=\bar{C}_k$, with
the following property: for each $\epsilon>0$ a weighted function $\mu(x),
0<\mu(x) \le1,| \{x\in[0,2\pi]: \mu(x)\not =1 \}| <\epsilon $ can be
constructed, so that the series is universal in the weighted space
$L_\mu^1[0,2\pi]$ with respect to rearrangements.
