The Germans did build two such submarines, named Deutschland and Bremen (Six more were converted into warships during their construction). They were constructed in 1916 by a private company, a subsidiary of the North German Lloyd shipping company. Bremen would sink on her maiden voyage, in August 1916. The cause of her loss is unknown, with some speculation that she was rammed by one of the auxiliary cruisers of the British 10th Cruiser Squadron - both Alsatian and Mantua reported collisions with submerged objects at about the right time. Deutschland was far more successful. For her first voyage, she departed Kiel on the 23rd June 1916, with a cargo of dyes and precious stones. She arrived at Baltimore on the 9th July - some footage of this arrival can be found here. She returned to Bremen on August 24th, carrying a load of strategic materials, including copper, zinc, nickel and rubber. She made another trip to New London in November, again carrying dyes on the route out, and strategic materials on the way back. Shortly before this voyage, the submarine U-53 visited Newport. U-53 was to have supported Bremen , but after her sinking, U-53s captain was ordered to make a visit to the American coast. This may have been an attempt to demonstrate to the US that German submarines could threaten the Eastern Seaboard, should the US enter the war. U-53 sank five ships off Nantucket, before returning to Germany. Following the entry of the United States into the war, Deutschland would be requisitioned by the navy and converted to a submarine-cruiser, as her long range and large size made her very suitable for such a role. She made several war patrols, sinking 43 ships and damaging a further 3 over the course of three patrols. One of these would take her back to the American coast, while the other two were to the region of the Azores. At the end of the war, she was seized by the Royal Navy as part of the armistice terms. After a thorough examination by the RN, she was opened to the public as an exhibition ship, touring the ports of the British south and east coasts.

Deutschland's mercantile voyages were the source of much legal wrangling. While merchants of belligerent powers were allowed to trade in neutral ports, it wasn't entirely clear how submarine merchants fit into this. Given that all other submarines of the time were armed, and it wasn't clear to the Allies that Deutschland wasn't, the Allies felt she should be treated as a warship. This view was bolstered by the fact that, especially when submerged, that she was almost impossible to tell from a warship. Such ships were legitimate targets, and could be sunk on sight. The Germans, of course, held that she was a legitimate merchant vessel, and subject to prize rules.

Sources:

The German Submarine War 1914-1918, R H Gibson, Maurice Prendergast, Constable & Co 1931, reprinted by Periscope Publishing, 2002

The Trading with the Enemy Act of 1917 and Synthetic Drugs, Dale Cooper, Pharmacy in History, Vol. 47, No. 2 (2005), pp. 47-61

VICTIM OR PARTICIPANT? ALLIED FISHING FLEETS AND U-BOAT ATTACKS IN WORLD WARS I AND II, found at http://www.cnrs-scrn.org/northern_mariner/vol01/tnm_1_4_1-18.pdf

Zimmermann Telegram: The Original Draft, Joachim von zur Gathen, Cryptologia, Vol 31, No. 1 (2007), published online at http://www.tandfonline.com/doi/full/10.1080/01611190600921165

Fighting the Great War at Sea: Strategy, Tactics and Technology, Norman Friedman, Seaforth, 2014