Try O-rings
 

If you pick up some O-rings, you can put several of them over the mic shaft, and on the XH-A1 space them so they fit between the rubber inside the mic mount. The right size o-rings will fit nicely, hold the mic firmly and look professional.

Not a bad idea... but did you notice that you posted this to the Sony Z1 forum? :-)

Sony HVR-Z1 / HDR-FX1 Shot gun Mount
 

I just bought the HVR-Z1 shotgun mic mount for my FX1. I hope it works like everyone says. I really need to slove this issue. I will be sure to upload photos and a full description of my experience. Thanks to everyone for your help.

Here is what I did
 

I took a clip on mic mount, the one with the butterfly bolt, uncrewed the mic mount, too my audio technica shock mount, unscrewed the hot shoe mount, screw the clip mount into the shock mount and mounted the whole thing on my camera handle, it works fine.

Where did you buy yours from?

I bought mine from here;

Sony FX1 Z1U Mic Microphone Holder kit ME66 AT897 Rode - eBay (item 170318996115 end time Jun-05-09 19:51:32 PDT)

i got this exact same mic holder. Transmits some noise from handling the cam, but overall I'm very happy with the product. I just wish that the adapter was black (mine was white colored).