Kathi, vital records are not and never have been kept at either the national, state or provincial level in Germany. In Germany, vital records are kept strictly at the local level. In order to proceed, you will thus first have to do additional research in the U.S. and find out exactly which city, town or village in Hesse-Darmstadt Ludwig Benner was from.
I should also add here that the state of Hesse-Darmstadt did not introduce the civil registration of births, deaths and marriages until 1876. That means that there is no birth record for Ludwig Benner. All that would be available would be his church baptismal record.
Here is the link to a map of the former Hesse-Darmstadt, so that you know what it looked like. (The former Hesse-Darmstadt, the capital of which was the city of Darmstadt, is the state in green.) To give you an idea of size as you look at the map, Hesse-Darmstadt covered an area about twice that covered by the U.S. state of Rhode Island. You will note that the region of Hesse-Darmstadt known as Upper Hesse (in German: Oberhessen; major city: Giessen) was separated from the southern portion of the state. I should also add here that not all of the former Hesse-Darmstadt is part of today's postwar German state of Hesse (in German: Hessen). That region of Hesse lying west of the Rhine River known as Rhenish Hesse (in German: Rheinhessen), which includes cities such as Mainz, Worms, Alzey, and Bingen, is part of today's postwar German state of Rhineland-Palatinate (in German: Rheinland-Pfalz). Ludwig Benner thus could have been from one of two of today's German states:http://www.gonschior.de/weimar/Hessen/uebersichtskarte.html