Hmmm... I think "remember" is normally understood to mean "successfully retrieve from memory," and quite rarely would mean "put in memory" (for which "memorize" is a better word). Admittedly, sometimes there could be ambiguity. I mean, if someone said "I will remember that," 99% of the time I would take the meaning to be "I will not forget that" or "I will not fail to retrieve it from memory" -- but probably not "I will carefully memorize that."
So "I will remember that" COULD in rare cases be translated with something like Я это запомню ("I shall memorize that") or Я это выучу наизусть ("I shall learn it by heart"), but usually it means Я это не забуду.