Background: Antibodies targeting calcitonin gene-related peptide (CGRP) or its receptor have shown efficacy in the prevention of migraine attacks. We investigated the efficacy and tolerability of fremanezumab, a fully humanised CGRP antibody, in patients with migraine who had previously not responded to two to four classes of migraine preventive medications. Methods: The randomised, double-blind, placebo-controlled, parallel-group, phase 3b FOCUS trial was done at 104 sites (including hospitals, medical centres, research institutes, and group practice clinics) across Belgium, the Czech Republic, Denmark, Finland, France, Germany, Italy, the Netherlands, Poland, Spain, Sweden, Switzerland, the UK, and the USA. We enrolled participants aged 18–70 years with episodic or chronic migraine who had documented failure to two to four classes of migraine preventive medications in the past 10 years. Failure was defined as no clinically meaningful improvement after at least 3 months of therapy at a stable dose, as per the treating physician's judgment; discontinuation because of adverse events that made treatment intolerable; or treatment contraindicated or unsuitable for the preventive treatment of migraine for the patient. Participants were randomly assigned (1:1:1) by electronic interactive response technology to subcutaneously administered quarterly fremanezumab (month 1, 675 mg; months 2 and 3: placebo), monthly fremanezumab (month 1: 225 mg in episodic migraine and 675 mg in chronic migraine; months 2 and 3: 225 mg in both migraine subgroups), or matched monthly placebo for 12 weeks. The primary outcome was mean change from baseline in the monthly average number of migraine days during the 12-week treatment period. This trial is registered with ClinicalTrials.gov, number NCT03308968, and is now completed. Findings: Between Nov 10, 2017, and July 6, 2018, 838 participants with episodic (329 [39%]) or chronic (509 [61%]) migraine were randomly assigned to placebo (n=279), quarterly fremanezumab (n=276), or monthly fremanezumab (n=283). Reductions from baseline in monthly average migraine days over 12 weeks were greater versus placebo (least-squares mean [LSM] change −0·6 [SE 0·3]) with quarterly fremanezumab (LSM change −3·7 [0·3]; LSM difference vs placebo −3·1 [95% CI −3·8 to −2·4]; p<0·0001) and with monthly fremanezumab (LSM change −4·1 [0·34]; LSM difference vs placebo −3·5 [−4·2 to −2·8]; p<0·0001). Adverse events were similar for placebo and fremanezumab. Serious adverse events were reported in four (1%) of 277 participants with placebo, two (<1%) of 276 with quarterly fremanezumab, and four (1%) of 285 with monthly fremanezumab. Interpretation: Fremanezumab was effective and well tolerated in patients with difficult-to-treat migraine who had previously not responded to up to four classes of migraine preventive medications. Funding: Teva Pharmaceuticals.