A Study of LY2605541 and Glargine and Exercise in Participants With Type 1 Diabetes

Inclusion Criteria: * Are males or females that have had a diagnosis of Type I Diabetes Mellitus (T1DM) for at least 12 months and are receiving multiple daily insulin injections. Total daily insulin dose \<1.2 units per kilogram per day (U/kg/day); daily basal dose \>0.2 U/kg/day * Female participants: are women of child-bearing potential who test negative for pregnancy at the time of enrollment based on a urine pregnancy test and agree to use a reliable method of birth control during the study * Have a body mass index (BMI) of 18 to 30 kilograms per meter squared (kg/m\^2), inclusive * Have a fasting c-peptide \<0.3 nanomoles per liter (nmol/L) * Have a hemoglobin A1c (HbA1c) \<9% at screening Participants with T1DM are eligible for enrollment in Part B of the study only if they meet all of the following criteria: * Have a maximal oxygen uptake (VO2 max) of ≥25 milliliters (mL) of oxygen per kilogram per minute (O2/kg/min) (for women) or ≥30 mL O2/kg/min (for men) * Perform regular physical cardiorespiratory activity to achieve an average total energy expenditure of ≥500 metabolic equivalent of task (MET)-minutes per week during the last 3 months prior to screening Exclusion Criteria: * Have known allergies to LY2605541, insulin glargine, related compounds or any components of the formulation * Have a history of or current cardiovascular, respiratory, hepatic, renal, gastrointestinal, endocrine (apart from T1DM), hematological, or neurological disorders capable of significantly altering the absorption, metabolism, or elimination of drugs; of constituting a risk when taking the study medication; or of interfering with the interpretation of data * Have had episodes of severe hypoglycemia in the past 6 months (severe hypoglycemia is defined as having neurological symptoms consistent with neuroglycopenia and having required assistance in treatment by a second party) * Have a history of hypoglycemia unawareness * Regular use or intended use of any over-the-counter or prescription medications or nutritional supplements that may affect blood glucose, the body's sensitivity to insulin, or that promote weight loss within 14 days prior to dosing * Have an average weekly alcohol intake that exceeds 21 units per week (males up to age 65) and 14 units per week (males over 65 and females), or are unwilling to comply with study requirements regarding alcohol consumption * Currently smokes \>5 cigarettes per day, or are unwilling to comply with study requirements regarding smoking or use of tobacco products * Have a hemoglobin level \<8.0 millimoles per liter (mmol/L) (male) or \<6.4 mmol/L (female) at screening * Are currently participating in a weight loss program or plan to do so during the course of the study * Are receiving chronic (lasting longer than 14 consecutive days) systemic glucocorticoid therapy (excluding topical, intra-articular, and intra-ocular preparations) or have received such therapy within the 4 weeks before dosing * Have fasting triglycerides \>400 milligrams per deciliter (mg/dL) (4.52 mmol/L) * Have previous history or family history of deep vein thrombosis