  We develop a method for the accurate reconstruction of non-bandlimited finite
rate of innovation signals on the sphere. For signals consisting of a finite
number of Dirac functions on the sphere, we develop an annihilating filter
based method for the accurate recovery of parameters of the Dirac functions
using a finite number of observations of the bandlimited signal. In comparison
to existing techniques, the proposed method enables more accurate
reconstruction primarily due to better conditioning of systems involved in the
recovery of parameters. For the recovery of $K$ Diracs on the sphere, the
proposed method requires samples of the signal bandlimited in the spherical
harmonic~(SH) domain at SH degree equal or greater than $ K + \sqrt{K +
\frac{1}{4}} - \frac{1}{2}$. In comparison to the existing state-of-the art
technique, the required bandlimit, and consequently the number of samples, of
the proposed method is the same or less. We also conduct numerical experiments
to demonstrate that the proposed technique is more accurate than the existing
methods by a factor of $10^{7}$ or more for $2 \le K\le 20$.
