theme_fieldgroup_fieldset by default to render field groups. The single argument to that function usually contains all the fields that belong to that field group (apart from a rendered representation of those fields). So if you could override how to theme your specific field group, you would be done.
One way is to override this function in your theme: add a function
foo_fieldgroup_fieldset to your theme (if your theme is called
foo). Note that this function is then used for all field groups. The function would have to decide on its own whether to show or hide empty fields. See Overriding themable output for more details.
A different way would be to use a template file for rendering field groups. You would have to implement
hook_theme for this to work (either in your theme or in a module). Advantage of this is, that preprocess functions are executed before rendering the field group. Preprocess functions allow you to add template suggestions, so you could have a custom template for one field group and a default one for others.
Also: have a look at the Theme developer module. It tells you for each item on a Drupal page which theme functions, templates and preprocess functions where involved in the rendering and how you could override the output.