Rubella, also known as "German measles", is usually classified as an acute infection. Your question is an interesting one, however, because as many as half of all rubella cases produce virtually no symptoms. However rubella does stimulate the body's immune system sufficiently to cause development of antibodies; people who have been infected with rubella generally develop an immunity to the virus, after which the virus is no longer found in their system. Hence it is not generally considered to be a chronic disease.
Rubella's symptoms, even in an acute case, are relatively mild, but the disease remains a concern because a woman becoming infected during the first trimester of a pregnancy is at nearly 100% risk of miscarriage, stillbirth, or delivering a child with severe birth defects.
There have been some reports of "chronic" rubella causing joint pain and juvenile arthritis, but according to the National Institute of Health, researchers doing controlled studies have not found any connection between chronic illness and either rubella or the rubella vaccine.