Scratch Disks

Make sure you have both read and write access to the external drive. It should be formatted Mac OS Extended.

also do not use a USB connection, use Firewire or ESATA

In addition to the above, if the drive is formatted properly and you have used the drive with more than one user account on your Mac, you may have tried to set the scratch disk location to a directory owned by the other account.

If you don't want to have to deal with permissions on the +external drive,+ you can open the drive in GetInfo (select the drive icon in the Finder, then do Command-I). In the resulting window, verify if the drive is set to "Ignore ownership on this volume"

Since I'm online right now, I'll jump in.

First thing you need to do is check what format the drive is. Select the drive icon in the Finder. Then do Command-I (Hold down the Command key, aka Apple key, and hit the letter I). The resulting window will tell you what format the drive is. If it's FAT32 or NTFS (which would be logical on a Windows pc) then you are going to have to reformat the drive to +Mac OS Extended+ in order to use it with Final Cut Express.

(Explanation - Mac OS X can read & write to FAT32 drives; read but not write to NTFS drives. Neither pc format is supported for Final Cut Express or Pro.)

But if you choose to reformat the drive, be sure to back up anything you want to keep, because reformatting will erase the entire drive.

How can I make sure I have read and write access to ext hard drive?

At the bottom of the Info window which Martin describes is a category called Sharing & Permissions. Your degree of access will be identified there ("Read & Write," "Read only," "No access," etc.) If you reformat the drive, you should have Read & Write access by default.