We propose a novel device capable of measuring relaxation times of viscoelastic fluids as small as 1 ms. In contrast to most rheometers, which by their very nature are concerned with producing viscometric or nearly-viscometric flows, here we make use of an elastic instability that occurs in the flow of viscoelastic fluids with curved streamlines. To calibrate the rheometer we combine simple scaling arguments with relaxation times obtained from first normal-stress difference data measured in a classical shear rheometer. As an additional check we also compare these relaxation times to those obtained from Zimm theory and good agreement is observed. Once calibrated, we show how the serpentine rheometer can be used to access smaller polymer concentrations and lower solvent viscosities where classical measurements become difficult or impossible to use due to inertial and/or resolution limitations. In the absence of calibration, the serpentine channel can still be a very useful comparative or index device.