  The Schramm-Loewner evolution (SLE_\kappa) is a candidate for the scaling
limit of random curves arising in two-dimensional critical phenomena. When
\kappa < 8, an instance of SLE_\kappa is a random planar curve with almost sure
Hausdorff dimension d = 1 + \kappa/8 < 2. This curve is conventionally
parametrized by its half plane capacity, rather than by any measure of its
d-dimensional volume. For \kappa < 8, we use a Doob-Meyer decomposition to
construct the unique (under mild assumptions) Markovian parametrization of
SLE_\kappa that transforms like a d-dimensional volume measure under conformal
maps. We prove that this parametrization is non-trivial (i.e., the curve is not
entirely traversed in zero time) for \kappa < 4(7 - \sqrt{33}) = 5.021 ....
