Using the Spitzer Space Telescope and Herschel Space Observatory, we have conducted a survey of IR galaxies in the field of the galaxy cluster AS1063 at z = 0.347, which is one of the most massive clusters known and a target of the Hubble Space Telescope (HST) Cluster Lensing and Supernova Survey with Hubble and the Frontier Field surveys. The Spitzer/MIPS 24 μm and Herschel/PACS and SPIRE images revealed that the core of AS1063 is surprisingly devoid of IR sources, showing only a few detectable sources within the central r ∼ 1‧. There is, however, one particularly bright source (2.3 mJy at 24 μm 106 mJy at 160 μm), which corresponds to a background galaxy at z = 0.61. The modest magnification factor (4.0×) implies that this galaxy is intrinsically IR luminous ({L}_{{I}{{R}}}=3.1× {10}¹¹ {{{L}}}_⊙ ). What is particularly interesting about this galaxy is that HST optical/near-IR images show a remarkably bright and large (1 kpc) clump at one edge of the disk. Our follow-up optical/near-IR spectroscopy shows Balmer (Hα-H8) and forbidden emission from this clump ([O II] λ3727, [O III] λλ4959,5007, [N II] λλ6548,6583), indicating that it is an H II region. The H II region appears to have formed in situ, as kinematically it is part of a rotating disk, and there is no evidence of nearby interacting galaxies. With an extinction correction of {A}_{{V}}=1.5 mag, the star formation rate of this giant H II region is ∼10 M_⊙ yr^-1, which is exceptionally large, even for high-redshift H II regions. Such a large and luminous H II region is often seen at z ∼ 2 but is quite rare in the nearby universe.