The process of domestication leads to major morphological and genetic changes, which in combination are known as domestication syndrome that differentiates crops from their wild ancestors. We characterized the genomic and phenotypic diversity of the South American grain amaranth Amaranthus caudatus, which has been cultivated for thousands of years and is one of the three native grain amaranths of South and Central America. Previously, several models of domestication were proposed including a domestication from the close relatives and putative ancestors A. hybridus or A. quitensis. To investigate the evolutionary relationship of A. caudatus and its two close relatives, we genotyped 119 amaranth accessions of the three species from the Andean region using genotyping-by-sequencing (GBS) and compared phenotypic variation in two domestication-related traits, seed size and seed color. The analysis of 9,485 SNPs revealed a strong genetic differentiation of cultivated A. caudatus from the relatives A. hybridus and A. quitensis. The two relatives did not cluster according to the species assigment but formed mixed groups according to their geographic origin in Ecuador and Peru, respectively. A. caudatus had a higher genetic diversity than its close relatives and shared a high proportion of polymorphisms with their wild relatives consistent with the absence of a strong bottleneck or a high level of recent gene flow. Genome sizes and seed sizes were not significantly different between A. caudatus and its relatives, although a genetically distinct group of A. caudatus from Bolivia had significantly larger seeds. We conclude that despite a long history of human cultivation and selection for white grain color, A. caudatus shows a weak genomic and phenotypic domestication syndrome and is an incompletely domesticated species because of weak selection or high levels of gene flow from its sympatric close undomesticated relatives that counteracted the fixation of key domestication traits.