  We have used the Palomar Testbed Interferometer to perform very high
precision differential astrometry on the 0.25 arcsecond separation binary star
HD 171779. In 70 minutes of observation we achieve a measurement uncertainty of
approximately 9 micro-arcseconds in one axis, consistent with theoretical
expectations. Night-to-night repeatability over four nights is at the level of
16 micro-arcseconds. This method of very-narrow-angle astrometry may be
extremely useful for searching for planets with masses as small as 0.5 Jupiter
Masses around a previously neglected class of stars -- so-called ``speckle
binaries.'' It will also provide measurements of stellar parameters such as
masses and distances, useful for constraining stellar models at the 10^-3
level.
