A Study in Participants With Type 1 Diabetes Mellitus

The purpose of this study is:

* To compare the blood sugar control of LY2605541 with insulin glargine after 78 weeks of treatment.
* To compare the rate of night-time low blood sugar episodes on LY2605541 with insulin glargine during 78 weeks of treatment.
* To compare the number of participants on LY2605541 reaching blood sugar targets without low blood sugar episodes at night to those taking insulin glargine after 78 weeks of treatment.
* To compare the rate of hypoglycemia episodes on LY2605541 with insulin glargine during 78 weeks of treatment.


Eligibility Criteria:
Inclusion Criteria:

* Have had diabetes mellitus for at least 1 year
* Have an hemoglobin A1c (HbA1c) value less than 12% according to the central laboratory at screening
* Have a body mass index (BMI) less than or equal to 35.0 kilograms per square meter (kg/m\^2)
* Have been treated for at least 90 days prior to screening with the following:

  * Insulin detemir, insulin glargine, or human insulin isophane suspension (NPH) insulin in combination with premeal insulin,
  * Self-mixed or premixed insulin regimens with any basal and bolus insulin combination administered at least twice daily, or
  * Continuous SC insulin infusion therapy
* This inclusion criterion applies to female participants:

  * Are not breastfeeding
  * Test negative for pregnancy at screening and randomization based on serum pregnancy tests
  * Do not intend to become pregnant during the study
  * Have practiced a reliable method of birth control (for example, use of oral contraceptives or levonorgestrel, diaphragms with contraceptive jelly, cervical caps with contraceptive jelly, condoms with contraceptive foam, intrauterine devices, partner with vasectomy, or abstinence) for at least 6 weeks prior to screening
  * Agree to continue to use a reliable method of birth control during the study, as determined by the investigator (and for 2 weeks following the last dose of study drug)
* Capable of and willing and desirous to do the following: adhere to a multiple daily injection regimen, inject insulin with a prefilled pen and perform Self-Monitored Blood Glucose (SMBG) and record keeping as required by this protocol, as determined by the investigator. Caregiver may be responsible for all of the above.

Exclusion Criteria:

* Are using twice-daily insulin glargine having been inadequately controlled on once-daily dosed glargine prior to screening
* Have excessive insulin resistance defined as having received a total daily dose of insulin greater than 1.5 units per kilogram (units/kg) at the time of randomization
* Receiving any oral or injectable medication (other than metformin for treatment of polycystic ovarian disease) intended for the treatment of diabetes mellitus other than insulins in the 90 days prior to screening
* Lipid-lowering medications:

  * Are using niacin preparations as a lipid-lowering medication and/or bile acid sequestrants within 90 days prior to screening or are using lipid-lowering medication at a dose that has not been stable for greater than or equal to 90 days prior to screening
  * If a participant has not been on a stable dose of lipid-lowering medication for greater than or equal to 90 days prior to screening, the site should wait to screen the participant. If the results of the screening laboratory tests require a change to the participant's current lipid-lowering medication or initiation of lipid-lowering medication, it is acceptable to change the lipid-lowering medication for the participant and to have the participant return greater than or equal to 90 days later to complete some of the screening procedures again
* Have fasting hypertriglyceridemia (defined as greater than 4.5 millimoles per liter \[mmol/L\], greater than 400 milligrams per deciliter \[mg/dL\]) at screening, as determined by the central laboratory
* Have had more than 1 episode of severe hypoglycemia (defined as requiring assistance due to neurologically disabling hypoglycemia as determined by the investigator) within 6 months prior to entry into the study
* Have had 2 or more emergency room visits or hospitalizations due to poor glucose control (hyperglycemia or diabetic ketoacidosis) in the past 6 months
* Cardiovascular: have cardiac disease with functional status that is New York Heart Association Class III or IV (per New York Heart Association Cardiac Disease Classification)
* Renal: Have a history of renal transplantation or are currently receiving renal dialysis or have serum creatinine greater than 2.5 mg/dL
* Hepatic: Have obvious clinical signs or symptoms of liver disease (excluding non-alcoholic fatty liver disease \[NAFLD\]), acute or chronic hepatitis, non-alcoholic steatohepatitis (NASH), or elevated liver enzyme measurements as indicated below:

  * Total bilirubin greater than or equal to 2 times the upper limit of normal (ULN) as defined by the central laboratory,
  * Alanine aminotransferase (ALT)/(serum glutamic pyruvic transaminase (SGPT) greater than 2.5 times ULN as defined by the central laboratory, or
  * Aspartate aminotransferase (AST)/serum glutamic oxaloacetic transaminase (SGOT) greater than 2.5 times ULN as defined by the central laboratory.
* Malignancy: Have active or untreated malignancy, have been in remission from clinically significant malignancy (other than basal cell or squamous cell skin cancer) for less than 5 years, or are at an increased risk for developing cancer or a recurrence of cancer in the opinion of the investigator
* Allergy: Have known hypersensitivity or allergy to any of the study insulins or their excipients
* Hematologic: Have had a blood transfusion or severe blood loss within 3 months prior to screening or have known hemoglobinopathy, hemolytic anemia, sickle cell anemia, or any other traits of hemoglobin abnormalities known to interfere with the HbA1c measurement
* Glucocorticoid therapy: Receiving chronic (lasting longer than 14 consecutive days) systemic glucocorticoid therapy (excluding topical , intraocular, intranasal, and inhaled preparations) or have received such therapy within 8 weeks immediately before screening with the exception of replacement therapy for adrenal insufficiency
* Diagnosed clinically significant diabetic autonomic neuropathy, in the opinion of the investigator
* Have any other condition (including known drug or alcohol abuse or psychiatric disorder including eating disorder) that precludes the participant from following and completing the protocol