  We study a scheme for implementing a controlled-Z (CZ) gate between two
neutral-atom qubits based on the Rydberg blockade mechanism in a manner that is
robust to errors caused by atomic motion. By employing adiabatic dressing of
the ground electronic state, we can protect the gate from decoherence due to
random phase errors that typically arise because of atomic thermal motion. In
addition, the adiabatic protocol allows for a Doppler-free configuration that
involves counterpropagating lasers in a $\sigma_+/\sigma_-$ orthogonal
polarization geometry that further reduces motional errors due to Doppler
shifts. The residual motional error is dominated by dipole-dipole forces acting
on doubly-excited Rydberg atoms when the blockade is imperfect. For reasonable
parameters, with qubits encoded into the clock states of $^{133}$Cs, we predict
that our protocol could produce a CZ gate in $<10$ $\mu$s with error
probability on the order of $10^{-3}$.
