The SU(2) part of electroweak gauge theory has sphaleronlike configurations even in its symmetric phase (temperature {ital T}{gt}{ital T}{sub {ital c}}) which mediate baryon-number-violating processes. These sphalerons sit on top of a potential barrier (which we construct explicitly) whose height rises linearly with {ital T}, and always exceeds {ital T} by a substantial factor. Such symmetric sphalerons are entirely nonperturbative, and we can at present only give lower bounds to the potential height, that is, the sphaleron mass {ital M}{sub {ital s}}. In terms of the Boltzmann factor exp({minus}{beta}{ital M}{sub {ital s}})==e{sup {minus}A}, when {ital T}{much gt}{ital T}{sub {ital c}}, {ital A} is a pure number independent of both {ital T} and {ital g}, the electroweak coupling constant. We estimate 13{lt}{ital A}{approx lt}40, corresponding to a Boltzmann factor between 2{times}10{sup {minus}6} and 4{times}10{sup {minus}18}. We do not discuss the full problem of small fluctuations around the sphaleron (necessary to find sphaleron-induced rates from the Boltzmann factor) but our explicitly constructed potential barrier gives a reasonable estimate of the single imaginary eigenvalue of small fluctuations. We also investigate high-temperature sphalerons in the presence of a finite baryon-number density, or equivalently a tachyonic Chern-Simons mass term. Such a term tends to reducemore » the sphaleron mass and increase the Boltzmann factor; the sphaleron never becomes tachyonic, no matter how large the expectation value of the Chern-Simons density.« less