If you lose weight and exercise can you get rid of the excess abdominal fat called an "apron? " or is a tummy tuck the only permanent solution?

Depends. It depends not on exercise, but more on the elasticity of your skin. If your skin has lost its elasticity, it won't automatically reshape around the smaller layer of fat, and you'd need an abdominoplasty to remove the redundant skin.
Not likely. If by the "apron" you mean a redundant collection of hanging skin, then the answer is no, that must be surgically removed. You should lose weight and exercise to get down to your goal weight and then seek out a plastic surgeon to begin the process of contouring your body.
Probably Not. Unfortunately, diet and exercise may not suffice when it comes to “getting rid” of excess skin of the lower abdominal wall. As a matter of fact, patients who have had significant weight loss may be left with significant redundant skin although very little adipose tissue is left. These patients often present for tummy tuck or panniculectomy surgery.
Apron is Fat plus. The "apron" is composed of fat and skin. It is also pushed forward and made more prominent by bulging and loose muscle underneath in most cases. Tummy tuck surgery addresses the big picture to reduce the problem more completely. Exercise can shrink the fat in there, but falls short in addressing the reminder of the problem.
Panniculectomy. The abdominal "apron" is called a pannus, and although it consists of fat and skin, diet and exercise reduces only the fat. The deflated skin apron will never retighten or go away without abdominoplasty or a lesser operation called panniculectomy. This operation may be covered by insurance but does not give the muscle tightening or much better results of a true tummy tuck!
Apron = Skin + Fat. Ususlly an abdominal apron or ""panniculus" is composed of skin and fat. Therefore while det and exercise may reduce fat and improve muscle tone, it is unlikely to make skin contract and, while highly desireable, may make the apron worse but easier to treat and better for your overall health. Loose skin is marked by stretch marks and possible intertrigo erythema and yeast infections.