  The entanglement in a quantum system that possess an internal symmetry,
characterized by the Sz-magnetization or U(1)-charge, is distributed among
different sectors. The aim of this letter is to gain a deeper understanding of
the contribution to the entanglement entropy in each of those sectors for the
ground state of conformal invariant critical one dimensional systems. We find
surprisingly that the entanglement entropy is equally distributed among the
different magnetization sectors. Its value is given by the standard area law
violating logarithmic term, that depends on the central charge c, minus a
double logarithmic correction related to the zero temperature susceptibility.
This result provides a new method to estimate simultaneously the central charge
c and the critical exponents of U(1)-symmetric quantum chains. The method is
numerically simple and gives precise results for the spin-1/2 quantum XXZ
chain. We also compute the probability distribution of the magnetization in
contiguous sublattices.
