  The differential multiplicity of dileptons in a hot and magnetized
quark-gluon plasma, $\Delta_{B}\equiv dN_{B}/d^{4}xd^{4}q$, is derived from
first principles. The constant magnetic field $B$ is assumed to be aligned in a
fixed spatial direction. It is shown that the anisotropy induced by the $B$
field is mainly reflected in the general structure of photon spectral density
function. This is related to the imaginary part of the vacuum polarization
tensor, $\mbox{Im}[\Pi^{\mu\nu}]$, which is derived in a first order
perturbative approximation. As expected, the final analytical expression for
$\Delta_{B}$ includes a trace over the product of a photonic part,
$\mbox{Im}[\Pi^{\mu\nu}]$, and a leptonic part, ${\cal{L}}_{\mu\nu}$. It is
shown that $\Delta_{B}$ consists of two parts, $\Delta_{B}^{\|}$ and
$\Delta_{B}^{\perp}$, arising from the components $(\mu,\nu)=(\|,\|)$ and
$(\mu,\nu)=(\perp,\perp)$ of $\mbox{Im}[\Pi^{\mu\nu}]$ and
${\cal{L}}_{\mu\nu}$. Here, the transverse and longitudinal directions are
defined with respect to the direction of the $B$ field. Combining
$\Delta_{B}^{\|}$ and $\Delta_{B}^{\perp}$, a novel anisotropy factor $\nu_{B}$
is introduced. Using the final analytical expression of $\Delta_{B}$, the
possible interplay between the temperature $T$ and the magnetic field strength
$eB$ on the ratio $\Delta_{B}/\Delta_{0}$ and $\nu_{B}$ is numerically studied.
Here, $\Delta_{0}$ is the Born approximated dilepton multiplicity in the
absence of external magnetic fields. It is, in particular, shown that for each
fixed $T$ and $B$, in the vicinity of certain threshold energies,
$\Delta_{B}\gg \Delta_{0}$ and $\Delta_{B}^{\perp}\gg \Delta_{B}^{\|}$. The
latter anisotropy may be interpreted as one of the microscopic sources of the
macroscopic anisotropies, reflecting themselves, e.g., in the elliptic
asymmetry factor $v_{2}$ of dileptons.
