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We consider the phenomenology of a class of gauge-mediated supersymmetry(SUSY) breaking (GMSB) models at a e+e- Linear Collider (LC) with c.o.m. energyup to 500 GeV. In particular, we refer to a high-luminosity (L ~ 3 x 10^34cm^-2 s^-1) machine, and use detailed simulation tools for a proposed detector.Among the GMSB-model building options, we define a simple framework and outlineits predictions at the LC, under the assumption that no SUSY signal is detectedat LEP or Tevatron. Our focus is on the case where a neutralino (N1) is thenext-to-lightest SUSY particle (NLSP), for which we determine the relevantregions of the GMSB parameter space. Many observables are calculated anddiscussed, including production cross sections, NLSP decay widths, branchingratios and distributions, for dominant and rare channels. We sketch how toextract the messenger and electroweak scale model parameters from a spectrummeasured via, e.g. threshold-scanning techniques. Several experimental methodsto measure the NLSP mass and lifetime are proposed and simulated in detail. Weshow that these methods can cover most of the lifetime range allowed byperturbativity requirements and suggested by cosmology in GMSB models. Also,they are relevant for any general low-energy SUSY breaking scenario. Values ofc*tau_N1 as short as 10's of microns and as long as 10's of metres can bemeasured with errors at the level of 10% or better after one year of LC runningwith high luminosity. We discuss how to determine a narrow range (<~ 5%) forthe fundamental SUSY breaking scale sqrt(F), based on the measured m_N1,c*tau_N1. Finally, we suggest how to optimise the LC detector performance forthis purpose.Comment: 56 pages, 32 figures (48 eps files), LaTeX + epsf.sty + colordvi.sty.  Revision v2: minor changes/additions, version to be published in EPJ

Measuring gauge-mediated supersymmetry breaking parameters at a 500 GeV $e^+e^-$ linear collider