  The free multiplicative Brownian motion $b_{t}$ is the large-$N$ limit of the
Brownian motion on $\mathsf{GL}(N;\mathbb{C}),$ in the sense of $\ast
$-distributions. The natural candidate for the large-$N$ limit of the empirical
distribution of eigenvalues is thus the Brown measure of $b_{t}$. In previous
work, the second and third authors showed that this Brown measure is supported
in the closure of a region $\Sigma_{t}$ that appeared work of Biane. In the
present paper, we compute the Brown measure completely. It has a continuous
density $W_{t}$ on $\bar{\Sigma}_{t},$ which is strictly positive and real
analytic on $\Sigma_{t}$. This density has a simple form in polar coordinates:
\[ W_{t}(r,\theta)=\frac{1}{r^{2}}w_{t}(\theta), \] where $w_{t}$ is an
analytic function determined by the geometry of the region $\Sigma_{t}$.
  We show also that the spectral measure of free unitary Brownian motion
$u_{t}$ is a "shadow" of the Brown measure of $b_{t}$, precisely mirroring the
relationship between Wigner's semicircle law and Ginibre's circular law. We
develop several new methods, based on stochastic differential equations and
PDE, to prove these results.
