Since it's not clear yet what might be causing your problem, don't start tearing drives out of your computer and replacing your Mint 13 (which you probably chose for your own good reasons) with a "newer" Mint (that, even if it were to boot quicker, would likely not reduce the boot time significantly in and of itself). Just because one drive is bigger than the other doesn't automatically mean that it is any faster, or the cause of your problem.
Eik wrote:The systems contains 2 HD : /dev/sda 200 GB /dev/sda1 bootable with XP and /dev/sdb 80 GB /sdb1 bootable with XP
Are you saying you have two separate Windows XP installations -- one on each drive?
From what you are saying, sounds like you have only XP-related partition(s) on /dev/sda and /dev/sdb is a mix of XP and Mint partitions. What is the /dev/sdb5 (NTFS) partition being used for? Did you designate that as your "/home" partition for Mint? Can you boot-up into your installed Mint 13 (not with live USB) and post back here with a screenshot of each hard drive from GParted please? Would like to see what is mounted when you're running your Mint install.
*** Menu -> Adminstration (or System) -> GParted
*** Wait 15 seconds or so for it to finish reading the drives
*** Take screenshot of GParted window showing /dev/sda; then using drop-down menu in upper-right of interface, select /dev/sdb and take another screenshot.
*** Post back here using the "Upload attachment" tab under the reply box in the forum interface.
I'm not entirely clear on what happens when you boot-up. If you have the computer set to boot from the first drive normally (/dev/sda), do you get any grub screen at all, or do you boot straight to Win XP? Do you only see a grub screen when forcing computer (with HP "bootloader") to boot instead from /dev/sdb?
Eik wrote:When I boot I get the following message : GNU GRUB version 1.99-21 Ubuntu 3
Minimal BASH-like line editing is supported. For the first word, TAB lists possible command comlplitions.
Are you saying it gets stuck with this message showing, or is that just something that displays for a few seconds before the grub menu listing boot choices shows up?