No, it was completely forgotten. The last tablets of the Gilgamesh epic to be produced date to the 2nd or 1st century BCE. The latest known references to Gilgamesh and/or its characters prior to 1872 are the following:

• The Book of Giants, a work of Jewish apocrypha found among the Dead Sea scrolls, refers to Gilgamesh and Humbaba, ca. 200 BCE. This book circulated among the Manichaeans for centuries afterward, so they would have known the names but probably not the original story.
• Roman author Claudius Aelianus (ca. 200 CE) incorrectly wrote in De Natura Animalium (12.21) that Gilgamesh was king of Babylon and successor to one "Seuechoros".
• Syrian theologian Theodore bar Konai (ca. 600 CE) mentioned "Gmigmos/Gligmos" as a king who was Abraham's contemporary.

None of these late references presuppose knowledge of the original Mesopotamian tale.

Source: Jeffrey H. Tigay, The Evolution of the Gilgamesh Epic, 2002, pp. 251ff

Update: The "Tale of Buluqiya", which is found in some versions of Arabian Nights, contains story motifs that might originate with the Epic of Gilgamesh.

See S. Dalley, “The Tale of Bulūqiyā and the Alexander Romance in Jewish and Sufi Mystical Circles,” in Tracing the Threads: Studies in the Vitality of Jewish Pseudepigrapha (1994), 239–69.