Flexible flat cable AWM 20798 female connector? (16 pin/1mm pitch)
I believe what you are looking for are called FFC, FPC (Flat Flexible) connectors.  It looks like DigiKey has quite a few at 0.039" (1.0mm) pitch and 16 conductor.

If you have a surface mount 1mm plug, this is what you need:
http://www.proto-advantage.com/store/product_info.php?products_id=3400022

If you use a through hole 1mm plug, you need something like this (much cheaper and easier):
http://www.proto-advantage.com/store/product_info.php?products_id=3800027

Never dealt with this company, so this isn't an endorsement.  You may find more by searching for Pitch Changer.