Yes, lack of 48-bit LBA support in the BIOS MAY be the root of this problem. If that is the case, you CAN use the 160 GB new drive IF you are careful about a couple of things. The key is to recognize that, in that situation where the mobo controller cannot handle any HDD larger than 137 GB (HDD maker's count) or 128 GB (by Windows' way of counting the same thing), you CANNOT try to use any part of the HDD above that boundary. So you CAN create and use a first Primary Partition of that size and not more with no difficulty. BUT you must not try to make another Partition out of the left-over space above that.
Now, this poses a problem for OP that you MAY be able to fix. VERY likely when you made your clone to the new drive, it did so to a Partition of about 148 GB, rather than 128, and that's why it is giving you trouble. IF you still have the original 80 GB old drive in good condition you can start over and re-make the clone to a Partition of the correct size. If you want to do this, proceed as follows.
FIRST is that you probably have some new files with info on that new drive already. This process will destroy all those! So, back up ANY files that are not already on the old drive. You will have to restore those to the new clone when this is all done.
Re-install the old HDD and get your cloning software ready. Read the software's manual, or maybe even start it up, and look through the menus to find a place to specify the size of the Partition being created on the new Destination drive to receive the clone copy.
Right here is where you will have to figure out which system they are using for specifying size? If it talks about not exceeding a limit on older drives of either 128 GB or 137 GB, that will answer the question. Or, if it tells you that your new drive is either 148 GB or 160 GB, that will answer. If your old HDD is ALL used as one volume, it may tell you its size is either 74 GB or 80 GB. NOTE these are NOT "Free Space", these are TOTAL disk capacity. So, can you tell whether the cloning software is using the smaller number system or the larger number system? If you can, you now know that the limit on the new disk's Partition size will be either 128 GB (Windows' view of the size) or 137 GB (Hard Drive maker's way of counting).
So, set the software options to limit the new Partition size to that. It may be, in fact, that the software will tell you that there is NO space available on the new 160 GB HDD to use (after all, you already have a clone copy on it). If that is the case and you KNOW you can make the right adjustments to get the re-cloning done properly, you may need to tell that software to delete all existing Partitions and data on the160 GB drive before proceeding to make a new one on the resulting empty disk.
So, if you can get all that arranged, you can make a new Primary Partition of up to 128 GB on the new HDD and place the clone copy there. Once that is done you can shut down and re-establish the recent configuration - no old HDD, and the new one with its clone copy as the boot device. It should boot with no trouble. If so, you can restore those copies of new files that you saved before starting.
If you do this, do NOT try to use Disk Management or any other tool to create a second Partition for more data in the Unallocated Space of the HDD. If you do that, there is a VERY good possibility that trying to put data there will really put data into the middle of files on the boot Partition instead, corrupting those files! The corruption could easily be in the file control system or the Windows system files, making the entire drive useless!