A Study of LY2605541 (Insulin Peglispro) and Human Insulin Concentrations in Fat Tissue

Inclusion Criteria: * Have a diagnosis of Type 1 Diabetes Mellitus (T1DM) based on medical history for at least 1 year prior to enrollment * Have a c-peptide value ≤0.3 nanomoles per liter (nmol/L) at screening * Have a serum creatinine value within normal limits at screening * Have a haemoglobin A1c (HbA1c) value ≤75 millimoles per mole (mmol/mol) (9.0%) at screening * Have a body mass index (BMI) of 20.0-30.0 kilograms per meter squared (kg/m\^2), inclusive, at screening Exclusion Criteria: * Have known or suspected allergies or hypersensitivities to LY2605541, human insulin, sinistrin, related compounds or any components of the formulations * Are women who are pregnant or lactating * Have an abnormal blood pressure for the population as determined by the investigator * Have renal insufficiency or major renal disorders * Have proliferative retinopathy or maculopathy * Have lipodystrophy * Have any wound healing disorder or are prone to keloid or hypertrophic scar formation * Have results of screening prothrombin time (PT) and international normalized ratio (INR) tests that are significantly prolonged * Have a fasting triglycerides value \> 4.52 millimoles per liter (mmol/L) (400 milligrams/deciliter (mg/dL)) * Are receiving chronic systemic or inhaled glucocorticoid or have received such therapy within the 4 weeks before dosing * Have a total daily insulin dose greater than 1.2 units per kilogram (U/kg) * 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 * Regular use or intended use of non-selective beta blockers * Regular use or intended use of monoamine oxidase (MAO) inhibitors * Are currently participating in a weight loss program or plan to do so during the course of the study * Are unwilling to avoid excessive sun exposure, steam baths, saunas, and swimming during the study. Sun cream should be used during sun bathing for the 6-month period following the study * Are unwilling to avoid extensive consumption of food containing inulin during the study and in the 48-hour period leading up to the clamp