A Study of Dulaglutide (LY2189265) in Participants With Type II Diabetes

Inclusion Criteria: * Have type 2 diabetes (based on the World Health Organization's \[WHO\] diagnostic criteria) * Have been treated with basal insulin glargine once daily with or without metformin for at least 3 months prior to screening * Doses of once daily insulin glargine and metformin (if taken) must be stable during the 3-month period prior to screening. Doses of metformin are considered stable if all prescribed doses during this period are in the range between the minimum required dose (≥1500 mg/day) and the maximum approved dose per the locally-approved label * Have an HbA1c value ≥7.0% and ≤10.5% as assessed by the central laboratory at screening * Require further insulin glargine dose increase at week 3 per the treat-to-target (TTT) algorithm based on the SMPG data collected during the prior week * Have stable weight (±5%) ≥3 months prior to screening * Have body mass index (BMI) ≤45 kilograms per square meter (kg/m\^2) at screening * Are able and willing to administer once weekly randomized therapy * Are females of childbearing potential who must: * Test negative for pregnancy at screening, based on a serum pregnancy test * Agree to use a reliable method of birth control * Not be breastfeeding Exclusion Criteria: * Have been treated with ANY other antihyperglycemia regimen, other than basal insulin glargine once daily with or without metformin, within the 3 months prior to screening or between screening and week 3 * Have a history of ≥1 episode of ketoacidosis or hyperosmolar state/coma * Have a history of hypoglycemia unawareness within the 6 months prior to screening * Have been treated with drugs that promote weight loss within the 3 months prior to screening or between screening and week 3 * Are receiving chronic (\>14 days) systemic glucocorticoid therapy or have received such therapy within the 4 weeks prior to screening or between screening and week 3 * Have had any of the following cardiovascular conditions within the 2 months prior to screening: acute myocardial infarction (MI), New York Heart Association (NYHA) Class III or Class IV heart failure, or cerebrovascular accident (stroke) * Have a known clinically significant gastric emptying abnormality or have undergone gastric bypass surgery or restrictive bariatric surgery * Have acute or chronic hepatitis, signs and symptoms of any other liver disease, or alanine aminotransferase (ALT) level \>2.5 times the upper limit of the reference range, as determined by the central laboratory * Have a history of chronic pancreatitis or acute idiopathic pancreatitis, or were diagnosed with any type of acute pancreatitis within the 3 months prior to screening * Have an estimated glomerular filtration rate (eGFR) \<30 milliliters/minute/1.73 square meter (mL/min/m\^2), calculated by the Chronic Kidney Disease-Epidemiology (CKD-EPI) equation, as determined by the central laboratory; for participants on metformin, have renal disease or renal dysfunction (for example, a serum creatinine ≥1.5 mg/deciliter \[dL\] \[male\] or ≥1.4 mg/dL \[female\] or eGFR \[CKD-EPI\] \<60 mL/min/1.73 m\^2) * Have evidence of a significant, uncontrolled endocrine abnormality * Have any self or family history of type 2A or type 2B multiple endocrine neoplasia (MEN 2A or 2B) in the absence of known C-cell hyperplasia * Have any self or family history of medullary C-cell hyperplasia, focal hyperplasia, or carcinoma (including sporadic, familial, or part of MEN 2A or 2B syndrome) * Have serum calcitonin ≥20 picograms/mL, as determined by the central laboratory * Have evidence of a significant, active autoimmune abnormality * Have any other condition not listed in this section that is a contraindication for use of insulin glargine, or, for participants using metformin, have a condition that is a contraindication for the use of metformin and would require metformin discontinuation per label * Have a history of transplanted organ * Have a history of active or untreated malignancy, or are in remission from a clinically significant malignancy during the 5 years prior to screening * Have a history of any other condition which, in the opinion of the investigator, may preclude the participants from following and completing the protocol * Have any hematologic condition that may interfere with HbA1c measurement (eg, hemolytic anemias, sickle-cell disease)