The astronomical unit was defined to be exactly 149 597 870 700 meters in 2012, so it is no longer a measured value. In fact, it never was a measured value. It was instead a computed value, and it was computed rather strangely.

Prior to 2012, the astronomical unit was defined as the distance between the center of the Sun at which a tiny particle in an unperturbed circular orbit about the Sun would yield a value of exactly 0.0172020985 for the Gaussian gravitational constant. That value corresponds to the distance at which a tiny particle in an unperturbed circular orbit about the Sun would have an orbital angular velocity of 0.0172020985 radians per solar day. That value corresponds to an orbital period of 365.256898 days. That value, now called a Gaussian year, was based on measurements available to Gauss of the length of a sidereal year. (The currently accepted value of the sidereal year is 365.256363004 days of 86400 seconds each.)

This outdated value in the length of a sidereal year was one of the reasons the astronomical unit was given a defined value in 2012. There were other reasons. One is that that definition made the concept of the astronomical unit a bit (more than a bit?) counter-intuitive. With that definition, uncertainty in the computed value of astronomical unit depended on the ability of solar system astronomers to estimate the Sun's gravitational parameter (conceptually, the product of the Newtonian gravitational constant and the Sun's mass; in practice, a quantity that could be estimated directly as a consequence of models used to generate ephemerides) and the ability to measure time.

The ability to measure time (currently about one part in 1016
) has far outpaced the ability to estimate the Sun's gravitational parameter (currently less than one part in 1010). This means that if the 2012 change had not happened, the uncertainty in the astronomical unit would depend on the ability estimate the Sun's gravitational parameter, about three parts in 1011, or about 0.000000003%. That's a lot better than the 0.0000001% precision asked about in the question.