  The paper deals with two inverse problems for Sturm--Liouville operator
$Ly=-y" +q(x)y$ on the finite interval $[0,\pi]$. The first one is the problem
of recovering of a potential by two spectra. We associate with this problem the
map $F:\, W^\theta_2\to l_B^\theta,\ F(\sigma) =\{s_k\}_1^\infty$, where
$W^\theta_2 = W^\theta_2[0,\pi]$ are Sobolev spaces with $\theta\geqslant 0$,
$\sigma=\int q$ is a primitive of the potential $q$ and $l_B^\theta$ are
special Hilbert spaces which we construct to place in the regularized spectral
data $\bold s = \{s_k\}_1^\infty$. The properties of the map $F$ are studied in
details. The main result is the theorem on uniform stability. It gives uniform
estimates from above and below of the norm of the difference $\|\sigma
-\sigma_1\|_\theta$ by the norm of the difference of the regularized spectral
data $\|\bold s -\bold s_1\|_\theta$ where the last norm is taken in
$l_B^\theta$. A similar result is obtained for the second inverse problem when
the potential is recovered by the spectral function of the operator $L$
generated by Dirichlet boundary conditions. The results are new for classical
case $q\in L_2$ which corresponds to the value $\theta =1$.
