This thesis describes the development of an astrometric facility at the Sydney University Stellar Interferometer (SUSI) with an aim to measure at high precision the relative astrometry of bright close binary stars and ultimately to detect the presence of exoplanets within those binary star systems through observations of the systems' perturbed motion. At the core of the facility is a new beam combiner that is phase-referenced to an existing primary beam combiner in the visible wavelength regime. The latter provides post-processed fringe-tracking information to the former for fringe stabilization and coherent integration of pre-recorded stellar fringes using newly developed data reduction software. Interference fringe packets of a binary star are recorded alternately; first the fringe packet of the primary, then the secondary, finally back to the primary again. The measurement of the fringe packet separation is facilitated by an air-filled differential delay line and a network of interferometer-based metrology systems. Characterizations and initial astronomical observations carried out with the dual beam combiner setup demonstrated for the first time the success of the dual-star phase-referencing technique in visible (ă1µm) wavelengths. The current astrometric precision is larger than 100µas while the long term astrometric accuracy is yet to be characterized. In a parallel development, a complementary observing method using only the primary beam combiner is also demonstrated in this thesis. Relative astrometry of binary stars up to "0.8 2 separation with this technique has been demonstrated to have precision of better than 100µas. A simple detection limit analysis based on a list of target binary stars estimates up to two exoplanet detections can be achieved with SUSI if the new astrometric facility attains precision of 10µas while the primary beam combiner operates at its designed peak performance. Finally, one new stellar companion was resolved and a preliminary astrometry for another suspected companion was estimated from the astronomical observation data collected throughout the course of this thesis.