Q: What is the telephone number that I can contact customer service to inquiry. How much it would cost monthly for cobra insurance. A: COBRA is a federal law, it is not an insurance company. There is no phone number to contact them directly. This law requires an employer if there is more than 20 employees to provide you with the continuation of the group health policy. The cost for the COBRA continuation is what you are currently paying, what your employer pays and 2% administrative fee. You will need to contact your previous employer to request a COBRA packet and they will be able to tell you the monthly cost. You can call us at 1-877-279-7959 for any additional information.