"Yes, this IS the complete KJV Bible!"
The performances are very good, and I enjoy the music and subtle sound effects.
Please note: This is a word-for-word reading of the KJV, made more entertaining by music, sound effects, and different voices for different speakers besides the narrator. If you want a real dramatization of selected stories from the Old and New Testaments, based on the KJV, I recommend Living Scriptures. Their audio programs are expensive and not available here; you have to go to their website (no, I don't work for them).
I don't know what problems may have existed with this title in the past, but both the Old and New Testaments are now included. I downloaded this title on 6-24-12 and imported it into iTunes for syncing to my iPod, which plugs into my car stereo. I spot checked to make sure the whole Bible was there, from Genesis 1 through the very last words of Revelation 22.
Now, you may encounter a problem with the numbering of the parts, depending on the app you're using. There are 11 parts. In iTunes, these showed up as 1, 10, 11, 2, 3, and so on. I fixed this by right-clicking on each part in iTunes, selecting Get Info, and changing the part names to Part 01, Part 02, and so on to Part 11. This caused them to display in the correct order. I also omitted the full title from the part names, making them just Part 01, etc., so that the number of each part would be easy to see on my car stereo display.