solemn

First recorded in 1275–1325; Middle English solem(p)ne, from Old French or directly from Late Latin sōlennis, sōlempnis, Latin sōlemnis, variant of sollemnis “consecrated, holy,” derivative of sollus “whole”

1. See grave².

The English solemn ultimately comes from the Latin adjective sollemnis “performed or celebrated according to correct religious forms.” Sollemnis has no secure etymology, but the Romans themselves thought that it came from the adjectives sollus “whole, complete” and a derivative adjective formed from the noun annus “year,” and therefore interpreted sollemnis as meaning “taking place every year, annual.”
In English, the extension of solemn from applying to rites, ceremonies, holy days, or oaths to nonreligious actions or feelings arose in the mid-15th century. One imagines early religious rites and ceremonies as being (like modern ones) serious and reverential affairs, and that the individuals performing or participating in them did so with a corresponding grave and serious demeanor. So it is not hard to see how the current sense of "grave, sober, or mirthless" developed: applying first to the people who participated in religious rites, and then losing the connection with the rites themselves.