False! The cell has already replicated its DNA (during S phase, "S" for "synthesis" phase) previously. Only after the DNA has been replicated, does mitosis begin.
Remember, the cell cycle is: Interphase --> Mitosis --> Interphase --> Mitosis .--> ...
And each interphase is itself broken into three stages:
Gap stage 1 (G1), Synthesis (S), and Gap stage 2 G2), and it's during the S stage of interphase that the DNA is replicated.
As a side note, cells which stop dividing (such as most neurons) leave the cell cycle in G1 and go into a stage called G0.