Will you stop sweating if you exercise in a cold enough room?
Apparently you can sweat even in a cold enough room, without even doing exercise! The main causes of sweating include, among exercise and high body temperature, stress and anxiety. This condition is called cold sweating. See this article:


  The body normally produces sweat as a way to help keep cool. Sweating normally occurs with exertion such as when exercising or in high temperatures. However, there are other reasons for sweating. Sweating can be triggered by fear or anxiety, and this is often referred to as a cold sweat. It comes on suddenly and results in cool, damp skin. It is the body's reaction to stress as part of the "fight or flight" response that helps us to react in a dangerous situation.


So, even if you just think about doing heavy exercise, you might get sweat in hands or feet. 

But cold sweating is not the only cause. It may also be caused by a more severe condition known as hyperhidrosis. See this article:


  Normally, your sweat glands produce perspiration that’s carried to the skin’s surface when the air temperature rises, you develop a fever, you’re exercising, or you’re feeling anxious, nervous, or under stress. When those factors are no longer an issue, the nerves that signal sweating are put on hold. For the 2% to 3% of the population who have hyperhidrosis, however, the sweat glands don't shut off. They sweat even when the circumstances don’t call for it: when they’re in air conditioning, or while they’re sitting and watching television. Some people even tell their doctors that they sweat in a swimming pool.


So, one might start sweating even in a cold enough room, if not because of body heat, then because of anxiety, hormones or medical conditions. Also, this condition does not work the way you are speculating. When the surrounding environment is very cold, the body prefers to shut off blood supply to skin than to cool it down through skin i.e. sensing cold environment, the body shuts off blood supply to skin and diverts it to core of the body. This is what causes more urination in cold environment and is called cold diuresis. Have a look at this answer for more information on this point.