Does a mini DVI to HDMI cable exist?
There are three ways you can convert from a mini DVI port to HDMI display:
Option 1: Mini-DVI to DVI adapter and then get a DVI to HDMI cable.
Option 2: Mini-DVI to HDMI adapter and than get an HDMI to HDMI extension cable because the adapter would be to short to connect to the external monitor.
Option 3: Mini-DVI to HDMI (male) cord. This is the best, because less elements are involved.

Option 1, 2 and 3 will produce the same quality of the video

Here’s a video with mini dvi to hdmi adapter with sound, mini dvi to hdmi converter.