We formulate the problem of determining the Hausdorf dimension, d j , of the AptIonian packing of circles as an eigenvalue problem of a linear integral equation. We show that solving a finite-dimensional approximation to this infinite-order matrix equation and extrapolating the results provides a fast algorithm for obtaining high-precision numerical estimates for d f . We find that d, = 1.305686729(10). This is consistent with the rigorously known bounds on d f . and improves the precision of L h e existing estimate by three orders of magnihlde.