A Study of LY2605541 Versus Insulin Glargine on Blood Sugar

Inclusion Criteria: All Participants * Are healthy males or participants with T1DM * Have a screening body mass index (BMI) of 20.0-29.9 kilograms per square meter (kg/m\^2) Healthy Participants ONLY * Are overtly healthy, as determined by medical history and physical examination * Have a fasting blood glucose \<108 milligrams/deciliter (mg/dL) (6.0 millimoles/liter \[mmol/L\]) at screening Participants with T1DM ONLY * Have a diagnosis of T1DM for at least 1 year based on medical history * Have a screening c-peptide \<0.5 nanograms/milliliter (ng/mL) * Have a hemoglobin A1c (HbA1c) of 6.0 to 9.0% at screening * Have had no episodes of severe hypoglycemia in the past 6 months Exclusion Criteria: All Participants * Have a hemoglobin level \<12.0 grams/deciliter (g/dL) * Are currently a smoker, used tobacco products on a regular basis in the 6 months prior to screening, or are intending to smoke during the study Healthy Participants ONLY • Regular use or intended use of over-the counter or prescription medication within 7 and 14 days, respectively, prior to dosing (apart from vitamin/mineral supplements or occasional use of acetaminophen or ibuprofen). Participants with T1DM ONLY * Regular use or intended use of any over-the-counter or prescription medications or nutritional supplements that affect blood glucose or the body's sensitivity to insulin or that promote weight loss within 14 days prior to dosing * Are receiving chronic (lasting longer than 14 consecutive days) systemic or inhaled glucocorticoid therapy (excluding topical, intra-articular, and intraocular preparations) or have received such therapy within the 4 weeks before dosing * Require a total daily insulin dose exceeding 100 units (U) * Have fasting triglycerides \>400 mg/dL