  The $A^1\Sigma^+_u$ - $X^1\Sigma^+_g$ UV spectrum of Mg$_2$ has been
investigated with high resolution employing Fourier-transform spectroscopy and
laser excitation. Computer simulation and fit of line positions to the
overlapping structures in the spectra yield precise transition frequencies.
Starting with the well characterized ground state $X^1\Sigma^+_g$ from former
work we derived excited energy levels and report on the evaluation of the
$A^1\Sigma^+_u$ excited state, which is found to interact with another
electronic state, which we identify as the lower part of the (1)$^1\Pi_u$
state. A coupled channels fit to the level energies of the upper state yields a
reliable potential energy curve for the $A^1\Sigma^+_u$ for the range of
vibrational levels 1 $\le$ $v'$ $\le$ 46. A potential energy curve for the
(1)$^1\Pi_u$ state is proposed, but the (1)$^1\Pi_u$ state is only
characterized by its coupling to the A state, and no direct transition to a
level of the (1)$^1\Pi_u$ could be uniquely identified due to the overlapping
spectral structures.
