Question:How heavy is Earth's moon?

I had always heard that the center of gravity of our Earth-moon system was basically on the surface of the earth.

But here's an interesting bit of data:

Houston reports the instant at which we leave the lunar sphere of
influence.  This means simply that despite the fact we are only
thirty-four thousand nautical miles from the moon, and still 
174,000 away from the earth, the earth's pull has become dominant,
and the mathematical equations now recognize that fact.

I quoted that from the video snapshot below; I don't remember in what book that info is highlighted.

The center of gravity of the earth-moon system is not near the surface of Earth.

Google Video: John Lear tells all part 4 of 4