Right now, you can get a master's degree in informatics. You can complete a fellowship in informatics. However, there is no established "Clinical Informatics" medical speciality recognized by the American Board of Medical Specialties (ABMS). That's about to change.
According to a story on InformationWeek Health Care:
A committee of the American Board of Medical Specialties (ABMS) on July 27 recommended that clinical informatics be defined as a subspecialty of the American Board of Preventive Medicine and the American Board of Pathology, the two specialty boards that cosponsored the proposal. The ABMS board of directors and the ABMS assembly are expected to approve the recommendation in late September.
...board-certified physicians... will be eligible to sit for the board exam in clinical informatics... physicians currently working in the clinical informatics field will not have to go through a training program to sit for the board exam... there will be a pathway to board certification that doesn't require training in a certified program if candidates have significant experience in the field...
The American Medical Informatics Association (AMIA) has been lobbying ABMS for official recognition of clinical informatics for several years.
"The formal approval of a medical subspecialty in clinical informatics will benefit the healthcare system by bringing demonstrated informatics expertise to bear on the critical information management, documentation, and patient-safety issues that are a challenge to today's patient care environments," said Edward Shortliffe, MD, president and CEO of AMIA. "AMIA played an important role in working with the American Board of Preventive Medicine and other certification bodies to help assure that physicians who acquire informatics skills will be recognized and recruited to further enhance their roles on the front lines of modern medicine."
You can read the AMIA document that outlines the clinical informatics specialty here (PDF).