The recently proposed loop representation, used previously to find exact solutions to the quantum constraints of general relativity, is here used to quantize linearized general relativity. The Fock space of graviton states and its associated algebra of observables are represented in terms of functionals of loops. The ``reality conditions'' are realized by an inner product that is chiral asymmetric, resulting in a chiral asymmetric ordering for the Hamiltonian and in an asymmetric description of the left and right handed gravitons. This chirally asymmetric formulation depends on a splitting of the linearized field into self-dual and anti-self dual parts rather than into positive and negative frequency parts; as the former, but not the latter, is meaningful away from flat backgrounds this is expected to be useful in connecting the nonperturbative theory to the linearized theory. The formalism depends on an arbitrary ``averaging'' function that controls certain divergences, but does not appear in the final physical quantities. Inspite of these somewhat unusual features, the loop quntization presented here is completely equivalent to the standard quantization of linearized gravity.

40 pages