Inclusion Criteria:  Written informed consent. Male or female between the ages of 18 and 80 years, inclusive, at Screening. Initial diagnosis of TED within 7 years prior to Screening. Proptosis ≥3 mm from baseline (prior to diagnosis of TED), as estimated by treating physician, and/or proptosis >3 mm above normal for race and gender. Participants must be euthyroid with the baseline disease under control or have mild hypo or hyperthyroidism (defined as free thyroxine and free triiodothyronine levels <50% above or below the normal limits) at Screening. Every effort should be made to correct the mild hypo- or hyperthyroidism promptly and to maintain the euthyroid state for the duration of the trial. Does not require immediate surgical ophthalmological intervention and is not planning corrective surgery/irradiation during the course of the trial. Diabetic participants must have HbA1c ≤8.0% at Screening. Participants with a history of IBD (ulcerative colitis or Crohn's disease) must be in clinical remission for at least 3 months, with no history of bowel surgery within 6 months prior to Screening and no planned surgery during the trial. Concomitant stable therapies for IBD without modifications in the 3 months prior to Screening are allowed. Women of childbearing potential (including those with an onset of menopause <2 years prior to Screening, non-therapy-induced amenorrhea for <12 months prior to Screening or not surgically sterile [absence of ovaries and/or uterus]) must have a negative serum pregnancy test at Screening and negative urine pregnancy tests at all protocol-specified time points (i.e., prior to each dose and throughout the participant's participation in the Follow-up Period); participants who are sexually active with a non-vasectomized male partner must agree to use 2 reliable forms of contraception during the trial, 1 of which is recommended to be hormonal, such as an oral contraceptive. Hormonal contraception must be started at least 1 full cycle prior to Baseline and continue for 180 days after the last dose of teprotumumab. Highly effective contraceptive methods (with a failure rate <1% per year), when used consistently and correctly, include implants, injectables, combination oral contraceptives, some intrauterine devices, tubal ligation, sexual abstinence or vasectomized partner. Willing and able to comply with the prescribed treatment protocol and evaluations for the duration of the trial.  Exclusion Criteria:  Decreased best-corrected visual acuity due to optic neuropathy, as defined by a decrease in vision of 2 lines on the Snellen chart, new visual field defect or color defect secondary to optic nerve involvement within the last 6 months. Corneal decompensation unresponsive to medical management. Decrease in proptosis of ≥2 mm in the study eye between Screening and Baseline. Prior orbital irradiation, orbital decompression or strabismus surgery. Planned eyelid surgery during the course of the trial. Alanine aminotransferase or aspartate aminotransferase >3 × the upper limit of normal or estimated glomerular filtration rate ≤30 mL/min/1.73m2 at Screening. Use of any steroid (intravenous [IV], oral, steroid eye drops) for the treatment of TED or other conditions within 3 weeks prior to Screening. Steroids cannot be initiated during the trial. Exceptions include topical and inhaled steroids, as well as steroids used to treat infusion reactions. Any treatment with rituximab (Rituxan® or MabThera®) within 12 months prior to the first infusion of teprotumumab or tocilizumab (Actemra® or Roactemra®) within 6 months prior to the first infusion of teprotumumab. Use of any other non-steroid immunosuppressive agent within 3 months prior to the first infusion of teprotumumab. Any previous treatment with teprotumumab, including previous enrollment in this trial or participation in a prior teprotumumab trial. Treatment with any mAb within 3 months prior to Screening. Identified pre-existing ophthalmic disease that, in the judgment of the Investigator, would preclude trial participation or complicate interpretation of trial results. Use of an investigational agent for any condition within 60 days or 5 half-lives, whichever is longer, prior to Screening or anticipated use during the course of the trial. Malignant condition in the past 5 years (except successfully treated basal/squamous cell carcinoma of the skin or cervical cancer in situ). Pregnant or lactating women. Current drug or alcohol abuse or history of either within the previous 2 years, in the opinion of the Investigator or as reported by the participant. Known hypersensitivity to any of the components of teprotumumab or prior hypersensitivity reactions to mAbs. Human immunodeficiency virus, untreated or positive viral load for hepatitis C or hepatitis B infections. Any other condition that, in the opinion of the Investigator, would preclude inclusion in the trial. After 150 participants with a CAS <3 at Baseline have been randomized, an additional exclusion criterion will apply: CAS <3 at Baseline.