It actually has nothing to do with having an experimental and control group.
A double dissociation refers to documenting two distinct patterns of impairment in two different groups or individuals, proving that two functions are neurologically distinct.
For example, someone with hippocampal damage will have trouble forming declarative memories (memories you are consciously aware of). If you repeatedly present them with a tone followed by an electric shock, and then play the tone and ask them what's about to happen, they won't be able to answer correctly. However, they are still capable of forming a conditioned aversion, which means that the tone will induce a feeling of fear in them even though they have no conscious memory of getting shocked. In fact, it will elicit just as much fear in a patient with hippocampal damage as in a normal control.
In contrast, someone with amygdala damage will have no trouble learning the tone-shock association - they'll figure it out as quickly as a normal control would, in fact. However, they won't be afraid of the tone. Even though they know full well that tone=shock (declaratively), and the shock is just as painful to them as it would be for a normal control, they can't form the unconscious association that the tone should be feared.
A control group is important to determine which functions are intact (as opposed to just 'less impaired'). But for a double dissociation, you need two kinds of patients with opposite impairments - not just one patient group with one function impaired.
To illustrate why one patient group wouldn't be enough, imagine you have a patient who fails second-order theory of mind tests ('I know you don't know that I know where you hid the candy') but passes first-order theory of mind ('I know you don't know where I hid the candy'). Does that mean that first-order and second-order theory of mind are distinct skills? No, because we have yet to find anyone who fails first-order theory of mind tests who doesn't also fail second-order theory of mind. Instead, it makes more sense to think of the two tests as measuring the same thing, with the first-order tests just being easier than the second-order tests.
If you were to damage the hippocampus in one group and leave the other group undamaged, you'd know the hippocampal damage caused their memory problems. But when they successfully develop a phobia of a tone paired with a shock, you wouldn't know if that's just an easier kind of memory to form, or if it's truly a separate memory system. You need the patient with the opposite pattern of deficits to prove that you really do have two separate systems in play.