  We present the results of a lattice study of the normalization constants and
second moments of the light-cone distribution amplitudes of longitudinally and
transversely polarized $\rho$ mesons. The calculation is performed using two
flavors of dynamical clover fermions at lattice spacings between
$0.060\,\text{fm}$ and $0.081\,\text{fm}$, different lattice volumes up to
$m_\pi L = 6.7$ and pion masses down to $m_\pi=150\,\text{MeV}$. Bare lattice
results are renormalized non-perturbatively using a variant of the RI'-MOM
scheme and converted to the $\overline{\text{MS}}$ scheme. The necessary
conversion coefficients, which are not available in the literature, are
calculated. The chiral extrapolation for the relevant decay constants is worked
out in detail. We obtain for the ratio of the tensor and vector coupling
constants $f_\rho^T/f_\rho^{\vphantom{T}} = 0.629(8)$ and the values of the
second Gegenbauer moments $a_2^\parallel = 0.132(27)$ and $a_2^\perp =
0.101(22)$ at the scale $\mu = 2\,\text{GeV}$ for the longitudinally and
transversely polarized $\rho$ mesons, respectively. The errors include the
statistical uncertainty and estimates of the systematics arising from
renormalization. Discretization errors cannot be estimated reliably and are not
included. In this calculation the possibility of $\rho\to\pi\pi$ decay at the
smaller pion masses is not taken into account.
