False. While an empire will imply a civilization, a civilization does not mean an empire. First, let us take a close look at the definition of civilization. Civilization, as defined by Merriam-Webster, is "having a (relatively) high level of culture and technological development." Most groups of people can fit under this frame, as they develop some form of pro-activity that not only helps the person (people) to adapt and survive under conditions of the environment, but also to be able to expand not only through population, but also with technological and class structural development. A civilization can fit many categories, from a small village with a structural development, but also an empire (as stated by the question). However, that is not to say that they are always the same thing. Empires, on the other hand, is extremely different from the average civilization that takes place. Empires typically suggest that one group of "civilized people", through conquest and politics, is able to take large swathes of land and to be able to control it extremely well. This implies that there must be: 1) a system in which the army can be controlled, 2) A large population to draw recruits for the army, 3) Structural farming that produces enough food for each person inside the city (empire), as well as enough food to lay aside in case a disaster occurs, and 4) Having an intricate design of some form of roadway and trade from others. Take, for example, a well-known empire that we can all agree upon, the Roman Empire. The Roman Empire fits the definition of a civilization, in that, in wherever they expanded (and were successful in doing so), they were able to integrate the different people groups in the area under a common rule, and to enforce those rules through regulators in the army. They were able to adapt not only their military, but also their way of life in many different climates and locations, which helped strengthen the security of the empire. Class structure played an important role in the Roman Empire, with the Emperor on the top and slaves at the bottom. Not only that, but in taking large swathes of land and being able to rule other groups of people with not only relative ease, but also with a steady hand, implies a strong, secure empire.
Through these examples, you can see that Even though empire implies civilization, civilization does not imply empire. So your answer is false, as the terms do not connect both ways.