This paper studies the rank-$1$ tensor completion problem for cubic tensors. First of all, we show that this problem is equivalent to a special rank-$1$ matrix recovery problem. When the tensor is strongly rank-$1$ completable, we show that the problem is equivalent to a rank-$1$ matrix completion problem and it can be solved by an iterative formula. For other cases, we propose both nuclear norm relaxation and moment relaxation methods for solving the resulting rank-$1$ matrix recovery problem. The nuclear norm relaxation sometimes returns a rank-$1$ tensor completion, while sometimes it does not. When it fails, we apply the moment hierarchy of semidefinite programming relaxations to solve the rank-$1$ matrix recovery problem. The moment hierarchy can always get a rank-$1$ tensor completion, or detect its nonexistence. Numerical experiments are shown to demonstrate the efficiency of these proposed methods.

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