A Study to Evaluate the Safety, Tolerability, Pharmacokinetics, and Pharmacodynamics of MK-1293 Compared With a Basal Insulin in Participants With Type 1 Diabetes (MK-1293-005)

Inclusion Criteria: * Has type 1 diabetes mellitus diagnosed at least 12 months before screening * Subject to investigator discretion, is on a stable dose of insulin for at least 2 weeks before screening * Has a total daily insulin dose \<=1.2 units/kg * Has a screening hemoglobin A1c \<9.5% * Has a body mass index \>18.0 and \<=30.0 kg/m\^2 * Has a weight \>=50 kg * Female participant of reproductive potential has a serum beta-human chorionic gonadotropin level consistent with the nongravid state and agrees to use (and/or have her partner use) 2 acceptable methods of birth control until 2 weeks after the last dose of study drug * Postmenopausal female participant is without menses for \>=1 year * Surgically sterile female participant status is post hysterectomy, oophorectomy, or tubal ligation * Has not used nicotine or nicotine-containing products for at least 3 months before study start or smokes less than 10 cigarettes per day and is willing to abstain during the trial Exclusion Criteria: * Has a history of clinically significant gastrointestinal, cardiovascular, hematological, hepatic, immunological, renal, respiratory, genitourinary, or major neurological abnormalities or disease * Has a history of clinically significant endocrine abnormalities or diseases except type 1 diabetes mellitus * Has had any severe hypoglycemic episodes associated with hypoglycemic seizures, comas, or unconsciousness within the past 3 months * Has a history of diabetic ketoacidosis within the past 6 months * Has a history of significant multiple or severe allergies, anaphylactic reaction, or significant intolerability to drugs or food * Has a history of hypersensitivity to pharmacologic insulins * Is positive for hepatitis B surface antigen, hepatitis C, or Human Immunodeficiency Virus * Has had major surgery or donated or lost 1 unit of blood within 4 weeks before screening * Unable to refrain from use of any medication or herbal remedy from 2 weeks prior to the first dose of study drug to until the posttrial visit. Some medications are permitted and may be discussed with the investigators * Vaccination within 12 weeks of start of study participation * Consumes \>3 glasses of alcoholic beverages per day. Participants consuming 4 glasses of alcoholic beverages may be enrolled at the discretion of the investigator. * Consumes \>6 servings of caffeinated beverages per day * Is a regular user of any illicit drugs or has a history of drug abuse (including alcohol) within approximately 1 year * Is on a carbohydrate-restricted diet (\<100 grams carbohydrate per day); participants who are on a carbohydrate-restricted diet may be included if they agree to a diet consisting of \>=100 grams of carbohydrate daily throughout the study * Has a personal or family history of hypercoagulability or thromboembolic disease * Has used systemic glucocorticoids within 3 months of screening or anticipates treatment with systemic glucocorticoids during study participation.