According to Wikipedia:

In 1894, the Compagnie Impériale des Chemins de Fer Éthiopiens was formed to build and operate a railroad across eastern Ethiopia from Addis Ababa to the port of Djibouti in what was at the time French Somaliland. When the company failed in 1906, the line from the coast in Djibouti ended at a place in the desert country of eastern Ethiopia where the town of Dire Dawa sprang up.

In 1908, the assets of the company were transferred to a new firm, the Compagnie de Chemin de Fer Franco-Ethiopien de Jibuti à Addis Abeba, which received a new concession to finish the line to Addis Ababa. After a year of wrangling with the previous financiers and their governments, construction began anew. By 1915 the line reached Akaki, only 23 kilometers from the capital, and two years later came all the way to Addis Ababa itself.

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I understand there is no longer passenger service between Addis and Dire Dawa. I received the impression that today tourists are strongly discouraged from traveling by train from Dire Dawa to Djibouti for security reasons.

For more information about the railroad, go to: