  We study the $\pm J$ three-dimensional Ising model with a spatially
uniaxially anisotropic bond randomness on the simple cubic lattice. The $\pm J$
random exchange is applied in the $xy$ planes, whereas in the z direction only
a ferromagnetic exchange is used. After sketching the phase diagram and
comparing it with the corresponding isotropic case, the system is studied, at
the ferromagnetic-paramagnetic transition line, using parallel tempering and a
convenient concentration of antiferromagnetic bonds ($p_z=0 ; p_{xy}=0.176$).
The numerical data point out clearly to a second-order
ferromagnetic-paramagnetic phase transition belonging in the same universality
class with the 3d random Ising model. The smooth finite-size behavior of the
effective exponents describing the peaks of the logarithmic derivatives of the
order parameter provides an accurate estimate of the critical exponent
$1/\nu=1.463(3)$ and a collapse analysis of magnetization data gives an
estimate $\beta/ \nu=0.516(7)$. These results, are in agreement with previous
studies and in particular with those of the isotropic $\pm J$ three-dimensional
Ising at the ferromagnetic-paramagnetic transition line, indicating the
irrelevance of the introduced anisotropy.
