A high-temperature plasma is created when an intense laser pulse is focused onto the surface of a solid. An ultrafast pulse of x-ray radiation is emitted from such a plasma when the laser pulse length is less than a picosecond. A high-speed streak camera detector was used to determine the duration of these x-ray pulses, and computer simulations of the plasmas agree with the experimental results. Scaling laws predict that brighter and more efficient x-ray sources will be obtained by the use of more intense laser pulses. These sources can be used for time-resolved x-ray scattering studies and for the development of x-ray lasers.