How to remove welcome screen and its screen option checkbox?

Clicking the screen options on dashboard, the drop down area has welcome checkbox. So even after hiding the welcome screen, if user chooses to display the welcome screen. It still shows.

Is there a way to hide this option completely and turn off the welcome screen?
Or otherwise, a way to remove the welcome content completely and replace it with something else?

For WordPress 3.5+, adding remove_action( 'welcome_panel', 'wp_welcome_panel' ); to your functions.php removes the Welcome Panel from the Dashboard as well as its Screen Options menu.

See http://codex.wordpress.org/Plugin_API/Action_Reference/welcome_panel.