This is a minor Romantic-era lullaby by the French composer and music critic Hippolyte Barbedette. It is musically distinctive for its adherence to the salon style, emphasizing a tender, rocking melodic line over a simple accompaniment.
The primary challenge is maintaining a consistent, gentle rocking motion without rushing the 6/8 rhythm.
Comparable difficulty and character — good candidates for what to learn next.
Berceuse 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 Berceuse lasts about 3 minutes.
Berceuse 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 Berceuse by Hippolyte Barbedette, Believe Me, If All those Endearing Young Charms by Irish Folk Music, "s'kommt ein Vogel geflogen" by Siegfried Ochs — see the full similar-pieces list on this page.