We are undertaking a program to measure the characteristics of the intracluster light (ICL; total flux, profile, color, and substructure) in a sample of 10 galaxy clusters with a range of cluster mass, morphology, and redshift. We present here the methods and results for the first cluster in that sample, A3888. We have identified an ICL component in A3888 in V and r that contains 13 A?5 of the total cluster light and extends to 700 h 70-1 kpc (a??0.3r200) from the center of the cluster. The ICL color in our smallest radial bin is V - r = 0.3 A? 0.1, similar to the central cluster elliptical galaxies. The ICL is redder than the galaxies at 400 h70-1 kpc <r <700 h 70-1 kpc, although the uncertainty in any one radial bin is high. Based on a comparison of V - r color with simple stellar models, the ICL contains a component that formed more than 7 Gyr ago (at z > 1) with a high-metallicity (1.0 ZaS? <ZICL a?? 2.5 ZaS?) and a more centralized component that contains stars formed within the past 5 Gyr (at z a?? 1). The profile of the ICL can be roughly fitted by a shallow exponential in the outer regions and a steeper exponential in the central region. We also find a concentration of diffuse light around a small group of galaxies 1.4 h70-1 Mpc from the center of the cluster. In addition, we find three low surface brightness features near the cluster center that are blue (V - r = 0.0) and contain a total flux of 0.1M*. Based on these observations and X-ray and galaxy morphology, we suggest that this cluster is entering a phase of significant merging of galaxy groups in the core, whereupon we expect the ICL fraction to grow significantly with the formation of a cD galaxy, as well as the infall of groups.