Jessica Burley, managing director of NatMags, will take over Southall's position until a permanent replacement is found.
Adcock said: "I'm desperately sorry to be leaving NatMags after 14 years, but especially sad to leave my role on Cosmo. However, I have been unwell for much of 2007, which unfortunately resulted in my contracting pneumonia.
"As a result I have decided, with 23 years in magazines and the media under my belt, that now is a good time to take a short break in my career, take time to recover fully and enjoy a rare opportunity to spend some time with my kids and husband."
But Adcock intends on returning to work as soon as possible, albeit in a different role.
Duncan Edwards, chief executive of NatMags, said: "I am excited to be announcing the appointment of Justine to head-up Cosmo, but at the same time I am disappointed to be losing Jan after so many years.
"[Jan] has been a wonderful colleague and contributor to the business. I do, however, respect her decision to step down."