  We construct an operator that measures the length of a curve in
four-dimensional Lorentzian vacuum quantum gravity. We work in a representation
in which a $SU(2)$ connection is diagonal and it is therefore surprising that
the operator obtained after regularization is densely defined, does not suffer
from factor ordering singularities and does not require any renormalization. We
show that the length operator admits self-adjoint extensions and compute part
of its spectrum which like its companions, the volume and area operators
already constructed in the literature, is purely discrete and roughly is
quantized in units of the Planck length. The length operator contains full and
direct information about all the components of the metric tensor which
faciliates the construction of a new type of weave states which approximate a
given classical 3-geometry.
