As many have said, it is not about being weak; as in lacking strength. I refer to the non-dominant hand as the off hand (although I shoot using both hands when I can). Most of us have sufficient strength in the non-dominant hand to shoot if we only had the coordination needed.
What we lack mostly is coordination with the hand that we never or rarely shoot with. It may also be weaker.
We gain coordination by practicing that skill which we want to increase. It is sometimes referred to as muscle recruitment, or building neural pathways.
If you usually practice for an hour, perhaps try splitting your practice time 50-50 between your hands. Remember that you are training your "off-hand" at a lower level of skill compared to your dominant hand. Practice basics until you are competent with that "off-hand". Proper stance, grip, finger position, sight alignment, breath control, trigger squeeze and follow through.
For defensive shooting, it may be assumed that you should practice shooting the "off-hand" with out any support from the other hand or arm. It may be assumed that you are shooting "off-hand" because the dominant hand has been incapacitated.
NRA Life Member - Orange Gunsite Member - NRA Certified Pistol Instructor
"When plunder becomes a way of life for a group of men living together in society,
they create for themselves in the course of time a legal system that authorizes it and a moral code that justifies it." Frederic Bastiat