You can use the in_category
function to test if the current post in the loop is in the specific category you want to display full content for.
For example
if ( in_category( 'my_category') ) :
the_content();
else :
the_excerpt();
endif;
For more information on the in_category
function see the WordPress codex at: http://codex.wordpress.org/Function_Reference/in_category