Quantum computers promise to increase substantially the efficiency of solving certain computationally demanding problems like searching atabases and factoring large integers. One of the greatest challenges ow is to implement basic quantum computational elements in a physical system and to demonstrate that they can be reliably controlled. Single spins in semiconductors, in particular associated with defect centers, are promising candidates for practical and scalable implementation of quantum computing even at room temperature . Such an implementation may also use the reliable and well known gate constructions from bulk nuclear magnetic resonance (NMR) quantum computing. This paper report an implementation of a quantum logical NOT and a conditional two-qubit gate (CROT) with single spins in a solid. As quantum bits a single electron spin and a single carbon thirteen nuclear spin of a single nitrogen vacancy defect center in diamond are used. The quantum state of the electron spin can be read out optically. Owing to long decoherence and relaxation times the systems meets the requirements of hardware for quantum computation. Density matrix tomography of the CROT gate shows that the achieved performance of the two bit conditional quantum gate is promising. The gate fidelity achieved in our experiments is up to 0.9, good enough to be used in quantum algorithms. Further on, the system may allow for solid-state room temperature quantum computation.