Solemnity keeps counters, including ice counters, from being put on Dark Depths as it enters the battlefield, since it's a land (C.R. 121.6), so Dark Depths's last ability will trigger immediately upon entering the battlefield (C.R. 603.8). Note, however, that—
nothing on Dark Depths allows that permanent to transform, which is a term with a special meaning in Magic (review C.R. 701.27), and
although that ability will trigger, it needs to go on the stack first; only after all players pass in a row while that ability is on the top of the stack can it resolve (C.R. 603.3, 116.4; C.R. 605.4a, an exception, doesn't apply to that ability because of C.R. 605.1b).
EDIT (Sep. 25, 2018): One rule was renumbered with Dominaria.
EDIT (Sep. 25, 2018): One rule was renumbered with Dominaria.