A conically-curved crystal x-ray spectrograph of a design proposed previously has been implemented, and combined with an x-ray streak camera to produce a streak spectrograph. The spectrograph design differs from more common curved crystal designs in that it collects a solid angle of x-rays and produces a line focus spectrum perpendicular to the axis from the source. Initial efforts using a mica crystal have produced a time-resolving spectrograph of especial interest to time-resolution of weak laser-plasma emission, having moderate spectral resolving power (λ/Aλ 600-1000), collection efficiency 50-100 times that of a comparably deployed planar crystal spectrograph and streak camera, and time resolution of 20-40 ps. The device is demonstrated in experiments time-resolving the temperature and density of thin aluminum signature layers isolated in spherical plastic targets.