Advice needed on approaching to Drupal 6 and jQuery Mobile
I personally don't recommend splitting the project in 2 sites but rather in 2 themes. If the project is slightly different (I assume you however use the same database) then try to customized it via theme (and theme functions).

I suggest you use the Omega/Fusion Theme which are responsive themes.
A responsive theme doesn't need a special redirection.

However if you need more customization in additional install Mobile Tools Module.

Then create a domain called m.domain.com in additional to your site domain.com and set inside the Mobile Tools Module based on a agent to switch to the theme needed.

There's indeed no need to install Drupal again.
You should ask your host to treat the m.domain.com as an alias for domain.com. That way it points to the same Drupal installation and the theme switch can take place.

If you don't have a mobile to test install Firefox addon: https://addons.mozilla.org/en-US/firefox/addon/user-agent-switcher/