  We obtain sample-path large deviations for a class of one-dimensional
stochastic differential equations with bounded drifts and heavy-tailed L\'evy
processes. These heavy-tailed L\'evy processes do not satisfy the exponential
integrability condition, which is a common restriction on the L\'evy processes
in existing large deviations contents. We further prove that the solution
processes satisfy a weak large deviation principle with a discrete rate
function and logarithmic speed. We also show that they do not satisfy the full
large deviation principle.
