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Magnetic reconnection is a fundamental plasma process. Recent theoretical studies and numerical simulations have suggested that electrons can be efficiently accelerated in contracting magnetic islands and when magnetic islands coalesce. Using data from the STEREO spacecraft, we report in this article the first observation of macroscopic magnetic-island coalescence and a possible splitting, and the associated electron acceleration. On 24 May 2010, two magnetic islands were observed by both the STEREO-A and STEREO-B spacecraft to propagate out along the current sheet behind a coronal mass ejection (CME) and merge. Electron acceleration to above 10 keV is inferred through the observation of a type-III-like radio burst. The acceleration process occurred at a macroscopic scale, likely during the merging of the two magnetic islands. Our observation of the magnetic-island coalescence is supported by a 2.5-D axisymmetric magnetohydrodynamic simulation of CME in which the merging of post-CME magnetic islands is clearly identified.