I have found a few other documents at the Archives:
The birth record of John (Jean) Birnbaum (1808).
The birth record of Margaretha Hansen (1822).
The birth record of John (Joannes) Birnbaum (1777).
The marriage record of J. Birnbaum and A. Weimerskirch (1805).
The marriage record of H. Birnbaum and S. Sibenaler (1774).
The death record of Catherine Court (1807).
The death record of John Birnbaum(1818).
The death record of Anna Weimerskirch (1831).
Johann Birnbaum was born the 15th of March 1808 in Rameldange (municipality of Niederanven).
Margaretha Hansen was born May 16th 1822 in Gonderange (municipality of Rodenbourg). Her parents were Paulus Hansen, a farmer, and Maria Hoffman.
Johann Birnbaum married Anna Weimerskirch the 30th of January 1805 in Niederanven. Johann Birnbaum was baptized April 24th 1777 in Greiveldange (Parrish of Lennange), he died April 5th 1818 in Rameldange. The parents of Johann Birnbaum were Henri Birnbaum, a weaver, and Susanne Sibenaler. His godfather was Johann Sibenaler, his godmother was Maria Freyberger. Anna Weimerskirch was born October 2nd (or 12th) 1782 in Rameldange, she died February 16th 1831 in Rameldange. The parents of Anna Weimerskirch were Jean Weimerskirch, a day labourer, and Catherine Court, a day labourer, (she died December 17th 1807 in Niederanven).
Henri Birnbaum married Susanne Sibenaler January 18th 1774 in Lennange.
Here is the text which certifies the baptism of John Birnbaum (1777):
24 Aprilii. Baptizatus fuit Joannes filius legitimus Henrici Byrenbaum et Susanna conjugum levantibus Joanni Sibenaler et Maria Freyberger ambo ew Greiveldingen et celibatu.
Here are the references of the documents:
EC 95, Niederanven, ITEM 3, N 1796-1890, 10 Pluviose An XIII, 30.1.1805, N 38, P 14-15, Birnbaum-Weimerskirch
EC 95, Niederanven, ITEM 3, N 1796-1890, 16.3.1808, N 20, P 6, Birnbaum Jean
EC 97, Niederanven, ITEM 2, D 1801-1890, 17.12.1807, N 37, P 13, Court Catherine
EC 97, Niederanven, ITEM 2, D 1801-1890, 5.4.1818, N 11, Birnbaum Jean
EC 97, Niederanven, ITEM 2, D 1801-1890, 17.2.1831, N 9, Weimerskirch Anna
EC 109, Rodenbourg, ITEM 3, 1796-1890, 17.5.1822, N 6, Hansen Margaretha
RP 112, Lenningen, VOL 3, M 1740-1789, 18.1.1774, Birnbaum-Sibenaler
RP 112, Lenningen, VOL 4, B 1764-1792, 24.4.1777, P 287 en haut, Birnbaum Joannes
It is possible to order printed copies of the documents at the National Archives.
It may take some time to find the records of the ancestors of Margaretha Hansen.
There are new transcriptions at the Genforum site, they should render well in the browser window.