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I haven't tried this exactly, but when shooting a stethoscope - which has a very odd-shaped chromed part (diaphragm), I used a huge piece of white cloth to surround it, leaving a space for the camera lens.

Last year I took photos of vintage cars and was unable to avoid reflections in the chromed parts.  Even when I managed to have myself and the camera out of the reflections, someone/something else would pop in.  I'm looking forward to the answer you get, and see if I can do something better next time, although this is a very uncontrolled condition (unlike shooting the stethoscope indoors).

Regards,
Adelaide

How big is the sphere?  Would it fit into a light tent, you can get some that have a small flap to poke the camera lens through?

Wow...I think that nails it...a bit of work but "lighting what the chrome sees" seems to be the best way to do this.

Peco, it's not truly a "sphere" but a 1960's VW hubcap (pre-VW emboss stamping), so about a foot and a half wide and its reflection is equal to a fisheye mirror.

Thanks folks. I knew this wasn't going to be easy and I truly appreciate your help.