A topological superconductor (TSC) is characterized by the topologically protected gapless surface state that is essentially an Andreev bound state consisting of Majorana fermions. While a TSC has not yet been discovered, the doped topological insulator ${\mathrm{Cu}}_{x}{\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$, which superconducts below $\ensuremath{\sim}3\text{ }\text{ }\mathrm{K}$, has been predicted to possess a topological superconducting state. We report that the point-contact spectra on the cleaved surface of superconducting ${\mathrm{Cu}}_{x}{\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ present a zero-bias conductance peak (ZBCP) which signifies unconventional superconductivity. Theoretical considerations of all possible superconducting states help us conclude that this ZBCP is due to Majorana fermions and gives evidence for a topological superconductivity in ${\mathrm{Cu}}_{x}{\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$. In addition, we found an unusual pseudogap that develops below $\ensuremath{\sim}20\text{ }\text{ }\mathrm{K}$ and coexists with the topological superconducting state.