Widget areas are the wrong tools for what you need. They are built to offer a choice. Breaking that would be very difficult … and hard to understand for the user.
Alternative: Use a custom setting, for example in wp-admin/options-general.php
where the tagline and the site title is. There is even a hook for such additions:
do_settings_sections('general');
First, wait for the action admin_init
, and then register a new settings field:
add_filter( 'admin_init', 'wpse_76943_register_setting' );
function wpse_76943_register_setting()
{
register_setting( 'general', 'footer_text', 'trim' );
add_settings_field(
'footer_text',
'Footer text',
'wpse_76943_print_input',
'general',
'default',
array ( 'label_for' => 'footer_text' )
);
}
Easy. Now we need a function to print that field. Use a classic text input field with a class large-text
to make it wide enough:
function wpse_76943_print_input()
{
$value = esc_attr( get_option( 'footer_text' ) );
print "<input type='text' value='$value' name='footer_text' id='footer_text'
class='large-text' />
<span class='description'>Text for the footer.</span>";
}

That’s all!
To use the footer text in your theme just check if there is a value, then print it:
if ( $footer_text = get_option( 'footer_text' ) )
print "<p>$footer_text</p>";
I have written a plugin doing almost the same thing, it is just more flexible: Public Contact Data (Download as ZIP archive).