I just finished all my setup+configuration inside a "catalog" folder and just successfully copied it all to the root of my public_html directory. My Presta Version is 1.4.4.1.
Some/most of this is a repeat of above posts but I see they date back in some cases 2009 so here's the summary of what I did:
1. Using file manager in CPanel on my server I made a compressed file of the inside of my "Catalog" folder (the entire Presta website I made started in "www.mysite/Catalog/") called "catalog.zip".
DO NOT just rightclick on "Catalog" folder and compress it. Doing this will create yet another "catalog" folder = exactly what you already have!!!! Go INTO/INSIDE the folder, then select all the sub-folders and files, and then right click - compress.
2. A zip file will appear now inside your current shop's root.
I moved that zip file holding my entire Presta shop to the root of my server's public_html.
You could download the zipfile, then upload it to your public_html root if you prefer.
3. Still in CPanel File Manager move to your root of public_html.
unzip/uncompress it from within your root of public_html.
Now you still have your working copy inside "www.yoursite/Catalog/" and ALSO the second copy to now get activated.
This leaves you some safety until you see all works well.
4. Still in CPanel File Manager in your new root version of your shop, go into the folder called "Config".
Download or edit online if you want the "settings.inc.php" changing:
from define('__PS_BASE_URI__', '/your-old-presta-subfolder-name/');
to
define('__PS_BASE_URI__', '/');
Save or overwrite the file.
5. Now try going into your admin back office through the root version www.mysite/admin/.
It should allow you inside BUT you're not done yet.
6. Now go to Preferences -->SEO & URLs. Scroll way down and make sure PS directory only shows /
7. Go to Tools ---> Generators and click on the "Generate htaccess" then "Generate robots.txt" to sync those files.
If you skip the last step your front office site will still point to the old locations and the robot text will as well.
8. Go into "Modules" and fix any payment areas that require your specific URLs if required.
9. Go to PayPal.com and/or Google.com if you use them as payment gateways and again revise their settings URLs to match your new one.
After testing your new rooted site enough you could delete the old one or maybe just set it to "catalog mode" in Preferences --> Products to make it there for backup but not usable for ordering.
I will delete mine after about 2 weeks of testing my root one.