We propose that the visible and x-ray emission associated with the cosmic gamma-ray burst GRB 970228 but following it by hours and days was produced by a weaker continuation of the processes which produced the gamma-ray burst itself. This hypothesis predicts an instantaneous spectrum F n~n 21͞2 , resulting from radiative cooling of synchrotron-emitting electrons, at frequencies from the infrared to x rays and higher. The limited data support this prediction. [S0031-9007(98) The gamma-ray burst GRB 970228 [1,2] has been observed after a delay of 8-17 hours in x rays [3] and of 17 hours in visible light [4]. This marks the first detection of emission at lower frequencies following the gamma-ray observation of a gamma-ray burst (GRB) and the first detection of any visible counterpart to a GRB. We consider possible delayed visible and x-ray emission mechanisms, and conclude that the activity by the source of the GRB continued at a much reduced intensity for at least a day. There are hints of such continued activity in other GRB, and future observations can decide if this is true of GRB in general. The observed simultaneous multiband spectrum of GRB 970228 agrees with the predictions of relativistic shock theory when the flux is integrated over a time longer than that required for a radiating electron to lose its energy.Several mechanisms for the continuing x-ray emission of GRB 970228 should be considered. The simplest possibility is that the relativistic particles required to explain a GRB will collide with a surrounding dilute medium. This process has been suggested [5] as the source of the gamma-ray emission itself. These models face, however, a serious problem. The observed duration of x-ray emission [3] is roughly 1000 times the reported gamma-ray duration [1], despite a ratio of only ϳ50 in the observed frequency n obs . Most models of this type predict a much steeper decrease in frequency as a function of time. A specific calculation [6] predicts, for example, a time scale of emission~n 25͞12 obs . One can consider, alternatively, models in which hot electrons cool, and emit lower energy photons. These models face the same problem. For example, a model [7] in which relativistic electrons radiate their energy in a constant magnetic field predicts a time scale~n 21͞2obs . Another alternative model of the x-ray emission, thermal bremsstrahlung (as in a supernova remnant), may also be excluded because the required power ϳ10 45 erg͞s (at a cosmological redshift of a few tenths) would require an unachievable particle density .10 10 cm 23 even if the maximum plausible mass of 1M Ø is radiating.Instead, we suggest that the observed brief intense gamma-ray emission of a GRB is only the "tip of an iceberg"; it emits gamma rays at a much lower level for time of order a day following (and perhaps preceding) the intense outburst. GRB detectors necessarily have high backgrounds because they must have very broad angular acceptance; these high backgrounds, lack of angular discrimination, and necessarily short integration tim...