This solution is going to be hard to explain without pictures so bear with me. Call any corner of the dodecahedron a. Call the three corners surrounding a to be b, c, and d. Call q the point at the center of triangle bcd. Call p the point at the center of the dodecahedron.
Lets call the center of a pentagon E, where B is one of the corners of the pentagon.
We now know the distance from B to P and we know the distance from E to B from problem 123. The volume of the dodecahedron is composed of 12 pyramids. 7.663119 Thanks to Mark Hoyle for some help with this problem.