How to access files on a drive from an older system, mounted in a new system?

I've recently built a new system, after a rather large physical injury was sustained by my previous system (a precarious balance, and gravity, were not a happy mix). Surprisingly the /home drive of that system appears to have more-or-less survived the trauma. However...

I decided to use a fresh drive for / (and swap) partition(s), and another fresh drive for the new /home. Now that's working, I decided to install the old /home drive (that I had assumed until now would be entirely dead and without capacity for use) into the new system to recover the files and data (so far as is possible).

At this point I've run into a snag: I have no idea how to go about this (with Windows it was relatively easy, the new drive would be the latest character of the alphabet, and go from there).

With 'disk utility' (System -> Administration -> Disk Utitlity) I've worked out which drive it is (/dev/sda) but clicking on 'mount' produces an error:

1: helper failed with:

mount: according to mtab, /dev/sdb1 is already mounted on /

mount failed

...if it is mounted on / I can't see it. I'm also moderately confused by the disk (device /dev/sda) being referred to as /dev/sdb1.

Any and all insights would be incredibly welcome (I've already voted for: Idea #9063: New internal hard drives default automount at Brainstorm).

Details (so far as I know them):

1. 40GB disk is / and swap,
2. 1.0 TB Samsung is /home
3. 1.0 TB Hitachi is from the old system (and was the old /home drive).

Output from sudo fdisk -l pasted below:

Disk /dev/sda: 1000.2 GB, 1000204886016 bytes
255 heads, 63 sectors/track, 121601 cylinders
Units = cylinders of 16065 * 512 = 8225280 bytes
Sector size (logical/physical): 512 bytes / 512 bytes
I/O size (minimum/optimal): 512 bytes / 512 bytes
Disk identifier: 0x000bef00

   Device Boot      Start         End      Blocks   Id  System
/dev/sda1               1      121601   976760001   83  Linux

Disk /dev/sdb: 40.0 GB, 40018599936 bytes
255 heads, 63 sectors/track, 4865 cylinders
Units = cylinders of 16065 * 512 = 8225280 bytes
Sector size (logical/physical): 512 bytes / 512 bytes
I/O size (minimum/optimal): 512 bytes / 512 bytes
Disk identifier: 0x00037652

   Device Boot      Start         End      Blocks   Id  System
/dev/sdb1   *           1        4742    38084608   83  Linux
/dev/sdb2            4742        4866      993281    5  Extended
/dev/sdb5            4742        4866      993280   82  Linux swap / Solaris

Disk /dev/sdc: 1000.2 GB, 1000204886016 bytes
255 heads, 63 sectors/track, 121601 cylinders
Units = cylinders of 16065 * 512 = 8225280 bytes
Sector size (logical/physical): 512 bytes / 512 bytes
I/O size (minimum/optimal): 512 bytes / 512 bytes
Disk identifier: 0x000e8d46

   Device Boot      Start         End      Blocks   Id  System
/dev/sdc1               1      121602   976760832   83  Linux

Edited in response to @Danny Staple's answer:

I ran the following:

udo mkdir /mnt/oldhome
sudo mount -t ext3 /dev/sda1 /mnt/oldhome

The first part works as expected, and creates the directory, the second part runs for some time and errors out with the following:

mount: wrong fs type, bad option, bad superblock on /dev/sda1,
       missing codepage or helper program, or other error
       In some cases useful info is found in syslog - try
       dmesg | tail  or so

I must confess that I'm beginning to believe that the SMART report, that suggests the disk is healthy with a 'few' bad sectors, may be a little inaccurate.

Edited, as requested by @Danny Staple (below), with the output from dmesg | tail:

david@morpheus:~$ dmesg | tail
[  192.008425]         72 03 11 04 00 00 00 0c 00 0a 80 00 00 00 00 00 
[  192.008444]         3a 34 18 97 
[  192.008452] sd 0:0:0:0: [sda] Add. Sense: Unrecovered read error - auto reallocate failed
[  192.008464] sd 0:0:0:0: [sda] CDB: Read(10): 28 00 3a 34 18 97 00 01 00 00
[  192.008482] end_request: I/O error, dev sda, sector 976492695
[  192.008511] JBD: Failed to read block at offset 264
[  192.008529] JBD: recovery failed
[  192.008536] EXT3-fs (sda1): 
[  192.008541] ata1: EH complete
[  192.008547] error loading journal

It's my sad duty to share the news of the untimely death of one 1.0 TB Hitachi hard drive, due, I'm assuming from the heart-stopping clicks in its last moments of life, mechanical damage sustained in a fall. It, and its many contents, will be sorely missed.

Unfortunately the data was not recoverable by any of the suggestions raised in this question, which leaves me in a slightly awkward position: I don't want to have a non-answered question, so I'll side with the community votes and accept @Danny Staple's answer, since it seemed the most promising suggestion (and, again, was the most community-rewarded answer), but I will note for late-comers in future that this question was not (really) resolved, so the solution offered by @Danny may, or may not, work for others.

Thank you all for your help, and suggestions.