A basic requirement for quantum information processing systems is the ability to completely control the state of a single qubit1, 2, 3, 4, 5, 6. For qubits based on electron spin, a universal single-qubit gate is realized by a rotation of the spin by any angle about an arbitrary axis. Driven, coherent Rabi oscillations between two spin states can be used to demonstrate control of the rotation angle. Ramsey interference, produced by two coherent spin rotations separated by a variable time delay, demonstrates control over the axis of rotation. Full quantum control of an electron spin in a quantum dot has previously been demonstrated using resonant radio-frequency pulses that require many spin precession periods7, 8, 9, 10. However, optical manipulation of the spin allows quantum control on a picosecond or femtosecond timescale11, 12, 13, 14, 15, 16, 17, 18, permitting an arbitrary rotation to be completed within one spin precession period6. Recent work in optical single-spin control has demonstrated the initialization of a spin state in a quantum dot19, 20, 21, 22, as well as the ultrafast manipulation of coherence in a largely unpolarized single-spin state17. Here we demonstrate complete coherent control over an initialized electron spin state in a quantum dot using picosecond optical pulses. First we vary the intensity of a single optical pulse to observe over six Rabi oscillations between the two spin states; then we apply two sequential pulses to observe high-contrast Ramsey interference. Such a two-pulse sequence realizes an arbitrary single-qubit gate completed on a picosecond timescale. Along with the spin initialization and final projective measurement of the spin state, these results demonstrate a complete set of all-optical single-qubit operations.