Didn't work.
For some more information.
-Internet is stable when my phone isn't connected.
-When I connect my phone it constantly resets and I can only use like 3-10 seconds of internet every reset. x_X
-I have connected my phone for more than 5min. (happened only twice/thrice)
Help me... any more things I could try? x_X
Oh, your phone resets, not your router? The way it sounded, your router was resetting when you connected your phone to it.
*So, this may be a little confusing, but I just didn't want to delete what I already typed. I was searching for how to factory reset your type of phone and instead, I got a crap load of posts about the same issue you are having with that phone. It doesn't look like it's just with the WiFi, it's just rebooting in general. Everybody is calling them junk. Here is one link, talk[dot]sonymobile[dot]com/message/144906. Just do a Google search for "Sony Ericsson WT19i Restarting". There is a chance this could be a defect and a much bigger issue. *
In that case:
1) I would make sure your software is updated to the latest version released.
> To check to see if you have a update, go to Settings -> About Phone -> System Updates (This is recommended if you are not using a rooted phone) also found this -> sonymobile[dot]com/gb/software/
2) Double check Router and Phone settings.
Router = WPA2-Personal Encryption (Recommended), G-Only (As long as you don't own any wireless B devices), make sure MAC filter is disabled (want to make sure this isn't the issue), and that other devices can access the internet without issues with these settings.
Phone = OS is updated to current release for your phone. Delete your current connection with your wireless and recreate after router settings are correct. Also, make sure it's not somehow trying to attach to a open network around you.
3) If the above doesn't work and you haven't already did it, a good old factory reset may help.
Backing Up your Phone:
Google Backup = Settings -> My Accounts -> Select your Gmail account -> Make sure all items are checked for syncing (if you use these items). Settings -> Privacy Settings -> Check "Back up my data" and "Automatic Restore".
Factory Restore = Having a hard time finding instructions. If I come across them, I'll post.