  We propose a definition of the Wilson loop operator in the N=1 beta-deformed
supersymmetric Yang-Mills theory. Although the operator is not BPS, it has a
finite expectation value at least up to order (g^2 N)^2. This does not happen
generally for a generic non-BPS Wilson loop whose expectation value is UV
divergent. For this reason we call this a near-BPS Wilson loop and conjecture
that its exact expectation value is finite. We derive the general form of the
boundary condition satisfied by the dual string worldsheet and find that it is
deformed. Finiteness of the expectation value of the Wilson loop, together with
some rather remarkable properties of the Lunin-Maldacena metric and the
B-field, fixes the boundary condition to be one which is characterized by the
vielbein of the deformed supergravity metric. The Wilson loop operators provide
natural candidates as dual descriptions to some of the existing D-brane
configurations in the Lunin-Maldacena background. We also construct the string
dual configuration for a near-1/4 BPS circular Wilson loop operator. The string
lies on a deformed three-sphere instead of a two-sphere as in the undeformed
case. The expectation value of the Wilson loop operator is computed using the
AdS/CFT correspondence and is found to be independent of the deformation. We
conjecture that the exact expectation value of the Wilson loop is given by the
same matrix model as in the undeformed case.
