mid-14c., solemne, solempne, "performed with due religious ceremony or reverence; sacred, devoted to religious observances," also, of a vow, etc., "made under religious sanction, binding," from Old French solempne, solemne (12c., Modern French solennel) and directly from Latin sollemnis, sollempnis "annual, established, religiously fixed, formal, ceremonial, traditional," a derivative of sollus "whole, unbroken, complete" (from PIE root *sol- "whole, well-kept"), though the etymology is uncertain for the -emnis.
"The explanation that Latin sollemnis was formed from sollus whole + annus year is not considered valid" [Barnhart], but some assimilation via folk-etymology is possible. One of de Vaan's sources offers *soll-epli- "with all (due) religious performances, with all due rites" and dissimilation on the last -l-.
For the tendency to unetymological (euphonic?) -p- between -m- and -n-, compare Middle English sompnearie "book of dreams," from Medieval Latin somnarius; sumpnour, a Middle English variant of sumnour "summoner."
In Middle English also "famous, important; imposing, grand," hence Chaucer's friar, a ful solempne man but a religious hypocrite. Without reference to religion, "marked by seriousness or earnestness," from late 14c. The sense of "fitted to inspire devout reflection" is from c. 1400. Related: Solemnly; solemness.