According to Venice, California historian Paul Tanck, what we now call the disco ball was invented by Arthur Reese, town decorator for Abbot Kinney, founder of Venice of America. Reese seems to have been on the job in Venice from the year of its inception, 1905, and some time in the first or possibly second decade of that century, he designed a many-faceted ball to hang from the ceiling of the famous Venice Ballroom. This postcard from 1910 may or may not represent what we are talking about here. The spheres seem to have some kind of reflective material on them, maybe, but they're not covered solidly.
Someone else says that mirror balls were used in Europe, to repel witches. So that would make them considerably older than the 1900s, by a long stretch.
Discoball 3 courtesy of Vagamundos
Discoball 4 courtesy of Duruk
Record Player courtesy of Couchia
via this Creative Commons license