Target-of-opportunity observations of the dwarf nova SS Cygni were made by the Extreme Ultraviolet Explorer (EUVE) satellite between 1993 August 17.1 and 23.6 UT (MJD 9216.6 and 9223.1). The observations cover the rise and plateau phases of an anomalous outburst which began on August 15.5 and which reached maximum at V similar to 8.3 on August 20.5. During the observations, the brightness of the source as measured by the Deep Survey instrument rose from similar to 0.015 counts s(-1) to similar to 5 counts s(-1). A delay of similar to 3 days was initially observed between the EUV and optical light curves, but this delay decreased to zero as the light curves approached maximum. During the interval August 17.3 to 23.6 UT, the 72-130 Angstrom flux as measured by the Short Wavelength Spectrometer rose from similar to 6 x 10(-13) to similar to 6.7 x 10(-11) ergs cm(-2) s(-1), while the 72-90 Angstrom to 90-130 Angstrom count ratio remained constant at 1.39 +/- 0.05. Parameterizing the EUV spectrum with a blackbody absorbed by a column of cold neutral material, we derive the luminosity L(bl) and fractional emitting area f = L(bl)/4 pi R(wd)(2) sigma T-4 of the boundary layer as a function of temperature kT. For kT = 30 eV, N-H similar to 4.4 x 10(19) cm(-2), L(bl) similar to 5 x 10(32) ergs s(-1) and f similar to 1 x 10(-4). For kT = 20 eV, N-H similar to 7.0 x 10(19) cm(-2), L(bl) similar to 2 x 10(33) ergs s(-1), and f similar to 3 x 10(-3). The optical through far-UV luminosity of the accretion disk in outburst is L(disk) similar to 3 x 10(34) ergs s(-1). Therefore, while simple theory predicts that the luminosities of the boundary layer and accretion disk should be comparable unless the white dwarf is rotating very rapidly, zeta = L(bl)/L(disk) similar to 1, we find that zeta less than or similar to 0.07. The most interesting and enigmatic aspect of these observations is the EUV spectrum, which evolved homologously over roughly two orders of magnitude in luminosity. The constancy of the EUV spectrum over such a wide range of brightness seems impossible to reconcile with the standard picture of boundary layer emission.