A Study of Baricitinib (LY3009104) in Participants With Moderate to Severe Atopic Dermatitis

Have participated in Study JAIW (NCT03435081), and meet specific completion requirements for that study, and do not meet any of the following Exclusions: * Have significant uncontrolled cerebro-cardiovascular (e.g., myocardial infarction \[MI\], unstable angina, unstable arterial hypertension, severe heart failure, or cerebrovascular accident), respiratory, hepatic, renal, gastrointestinal, endocrine, hematologic, neuropsychiatric disorders, or abnormal laboratory values that developed during a previous baricitinib study that, in the opinion of the investigator, pose an unacceptable risk to the participant if investigational product continues to be administered. * Have a known hypersensitivity to baricitinib or any component of this investigational product. * Had investigational product permanently discontinued at any time during a previous baricitinib study, except for participants who had investigational product discontinued during originating study because of rescue with an oral systemic AD therapy (e.g., corticosteroid, cyclosporine, methotrexate). * Had temporary investigational product interruption continue at the final study visit of a previous baricitinib study and, in the opinion of the investigator, this poses an unacceptable risk for the participant's participation in the study. * Pregnant or breastfeeding OR • Have not participated in a Study JAIW (NCT03435081) and satisfy the following criteria: Inclusion Criteria: * Have a diagnosis of atopic dermatitis (AD) at least 12 months before screening. * Have moderate to severe AD, including all of the following: * EASI score ≥16 * IGA score of ≥3 * 10%- 50% BSA involvement * Have had inadequate response or intolerance to existing topical (applied to the skin) medications within 6 months preceding screening. * Are willing to discontinue certain treatments for eczema (such as systemic and topical treatments) * Agree to use emollients daily. Exclusion Criteria: * Are currently experiencing or have a history of other concomitant skin conditions (e.g., psoriasis or lupus erythematosus), or a history of erythrodermic, refractory, or unstable skin disease that requires frequent hospitalizations and/or intravenous treatment for skin infections. * A history of eczema herpeticum within 12 months, and/or a history of 2 or more episodes of eczema herpeticum in the past. * Participants who are currently experiencing a skin infection that requires treatment, or is currently being treated, with topical or systemic antibiotics. * Have any serious illness that is anticipated to require the use of systemic corticosteroids or otherwise interfere with study participation or require active frequent monitoring (e.g., unstable chronic asthma). * Have been treated with the following therapies: * monoclonal antibody for less than 5 half-lives before randomization * received prior treatment with any oral Janus kinase (JAK) inhibitor less than 4 weeks before randomization * received any parenteral corticosteroid administered by intramuscular or intravenous injection within 6 weeks of planned randomization or are anticipated to require parenteral injection of corticosteroids during the study * have had an intra-articular corticosteroid injection within 6 weeks of planned randomization * probenecid at the time of randomization that cannot be discontinued for the duration of the study * Have high blood pressure characterized by a repeated systolic blood pressure \>160 millimeters of mercury (mm Hg) or diastolic blood pressure \>100 mm Hg. * Have had major surgery within the past eight weeks or are planning major surgery during the study. * Have experienced any of the following within 12 weeks of screening: myocardial infarction (MI), unstable ischemic heart disease, stroke, or New York Heart Association Stage III/IV heart failure. * Have a history of venous thromboembolic event (VTE), or are considered at high risk for VTE. * Have a history or presence of cardiovascular, respiratory, hepatic, gastrointestinal, endocrine, hematological, neurological, lymphoproliferative disease or neuropsychiatric disorders or any other serious and/or unstable illness. * Have a current or recent clinically serious viral, bacterial, fungal, or parasitic infection including herpes zoster, tuberculosis. * Have specific laboratory abnormalities. * Have received certain treatments that are contraindicated. * Pregnant or breastfeeding.