This is a salon-style lullaby composed in the late 19th-century French tradition. It is distinguished by its gentle, rocking accompaniment and simple, vocal-like melodic phrasing.
Focus on maintaining a consistent, swaying rhythm in the left hand while projecting a singing tone in the right.
Comparable difficulty and character — good candidates for what to learn next.
Berce-moi ma très chère is rated Henle level 3 of 9 — late beginner repertoire, roughly ABRSM grades 4–5. In the RCM system it corresponds to about level 4.
You should be comfortable at Henle level 3 — meaning you can already play level 2–3 pieces cleanly. Learning it one level early is possible as a stretch piece, but more than that usually leads to months of frustration.
A typical performance of Berce-moi ma très chère lasts about 3 minutes.
Berce-moi ma très chère is available as free public-domain sheet music on IMSLP — the score is linked directly from this page.
Pieces at a comparable difficulty and character include Albumblad by Oskar Merikanto, Berceuse by Adolf Gutmann, Berceuse by Georges Marty — see the full similar-pieces list on this page.