Summary Background Whole-breast irradiation after breast-conserving surgery for patients with early-stage breast cancer decreases ipsilateral breast-tumour recurrence (IBTR), yielding comparable results to mastectomy. It is unknown whether accelerated partial breast irradiation (APBI) to only the tumour-bearing quadrant, which shortens treatment duration, is equally effective. In our trial, we investigated whether APBI provides equivalent local tumour control after lumpectomy compared with whole-breast irradiation. Methods We did this randomised, phase 3, equivalence trial (NSABP B-39/RTOG 0413) in 154 clinical centres in the USA, Canada, Ireland, and Israel. Adult women (>18 years) with early-stage (0, I, or II; no evidence of distant metastases, but up to three axillary nodes could be positive) breast cancer (tumour size ≤3 cm; including all histologies and multifocal breast cancers), who had had lumpectomy with negative (ie, no detectable cancer cells) surgical margins, were randomly assigned (1:1) using a biased-coin-based minimisation algorithm to receive either whole-breast irradiation (whole-breast irradiation group) or APBI (APBI group). Whole-breast irradiation was delivered in 25 daily fractions of 50 Gy over 5 weeks, with or without a supplemental boost to the tumour bed, and APBI was delivered as 34 Gy of brachytherapy or 38·5 Gy of external bream radiation therapy in 10 fractions, over 5 treatment days within an 8-day period. Randomisation was stratified by disease stage, menopausal status, hormone-receptor status, and intention to receive chemotherapy. Patients, investigators, and statisticians could not be masked to treatment allocation. The primary outcome of invasive and non-invasive IBTR as a first recurrence was analysed in the intention-to-treat population, excluding those patients who were lost to follow-up, with an equivalency test on the basis of a 50% margin increase in the hazard ratio (90% CI for the observed HR between 0·667 and 1·5 for equivalence) and a Cox proportional hazard ...