  The exclusive semileptonic $B\to \pi(\rho) e\nu$ decays are studied in the
framework of the relativistic quark model based on the quasipotential approach
in quantum field theory. The large recoil momentum of the final $\pi(\rho)$
meson allows for the expansion of the decay form factors at $q^2=0$ in the
inverse powers of $b$-quark mass. This considerably simplifies the analysis of
these decays. The $1/m_b$ expansion is carried out up to the first order. It is
found that the $q^2$-dependence of the axial form factor $A_1$ is different
from the $q^2$-behaviour of other form factors. We find $\Gamma(B\to \pi
e\nu)=(3.1\pm 0.6)\times \vert V_{ub}\vert ^2\times 10^{12}$s${}^{-1}$ and
$\Gamma(B\to \rho e\nu)=(5.7\pm 1.2)\times \vert V_{ub}\vert ^2\times
10^{12}$s${}^{-1}$. The relation between semileptonic and rare radiative
$B$-decays is discussed.
