Installation on large disc takes too long

You don't need to fill the drive at partition time. If it is e2fsck or mkfs causing problems, try installing smaller partitions. You can, for example, create a partition for "/" which is <20GB -- this will likely hold most of your system data. (I use a / partition of 4GB, and it is a bit too much of a squish, but it works). You can create a separate small /home partition, and a final /media/stuff partition later.

Oh, if you use ext4, its supposed to be much faster at filesystem creation and checking:
http://www.ep.ph.bham.ac.uk/general/support/raid/raidperf11.html

Your installation of two Ubuntu will not work properly. Both Ubuntus must share a dedicated partition to "/boot" folder. So you need to have four partitions. For maintainability issues, I advise you to create fourth one a extended, so later you could add more partitions if needed.
Why do you want two have two Ubuntus on computer ? Perhaps there is a easier solution.

I finally gave up on the repartitioning of the hard drive. I aborted the installation process and went straight to gparted to delete all partitions on the drive (I still have the USB drive with the backup of my data.) I then tried the installation process again using the LiveCD. I got through step 3 (specifying a keyboard layout) and once again the installation process hung up - for 24 hours. I don't remember this happening when I first installed Ubuntu on the drive, but it was a long time ago and I'm not good about taking and/or keeping notes. I now have a LiveCD and a /dev/sdb with 931.51 GiB of unallocated space. No partitions. What is the quickest way to get an Ubuntu installation on this drive and what are your recommendations for a partitioning scheme?

delance, I don't understand why you think the two partition/two Ubuntu scheme would not work. I was going to setup two completely separate Ubuntu file systems, one on each partition. Why would that not work. If can setup two different operating systems in that manner, as I have done on my laptop (one Windows and one Ubuntu.) That scheme works just fine so why would two totally independent Ubuntu setups not work? I was doing this so that I could mount the old Ubuntu file system under /media/oldubuntu on the new Ubuntu system and easily move data files to the new system. I had it in my mind to convert the old Ubuntu partition to some other operating system at a later date.

At start of disk (the 32 first KiloBytes) you have the MBR of Grub2, and it will point to one of the /boot directory. Each time you update the grub configuration, which means each time you change your kernel from one of the Ubuntu, the MBR could point to another /boot directory. So I'm not sure of consequences, and you could get a big mess.
When you update Grub, it search on all partitions the bootable software, so it could find both /boot directory, and propose to you both Ubuntu in Grub menu.
I never used such configuration, so I'm interested to know what will happen!
To avoid this, you could simply remove "/boot" directory of old Ubuntu, preventing Ubuntu to detect kernel images, and so you will boot only on new Ubuntu.

For your hung up, I've currently no idea. While in Keyboard page, do you see the copy progress bar increasing ?
And did you check ISO image: https://help.ubuntu.com/community/HowToMD5SUM

Correction: GRUB only reinstalls its loader to the boot track when you upgrade grub-pc. Kernel changes only affect /boot/grub/grub.cfg, not the boot track.

I would not recommend removing /boot; that will confuse a number of things, and in any event kernel or GRUB upgrades will simply put it back. If you don't want Ubuntu to detect other operating systems on the disk and include them in the boot menu, it's better to set GRUB_DISABLE_OS_PROBER=true in /etc/default/grub.

Gave Up!
Switched to "Linux From Scratch". Used cfdisk to setup partition table and it took about five minutes. Used mke2fs to setup filesystems and that took about five minutes. New partitions mounted just fine. I am now up to section 5.9 in the LFS v6.7 and everything is going fine.
I still have Ubuntu on my laptop and in the future I will attempt to install it on my desktop in the unallocated space I have left (I only used about 110 Gig for LFS.)
I just had to get going on something.

@Colin Watson: do you have documentation about the internals of GRUB ? I'm not really interested to read source code to understand how it works.