We present results of an intensive two-month campaign of approximately daily spectrophotometric monitoring of the narrow-line Seyfert 1 galaxy Akn 564 with the Hubble Space Telescope ( HST ) . The fractional variability amplitude of the continuum variations between 1365–3000 Å is \sim 6 % , about a factor 3 less than that found in typical Seyfert 1 galaxies over a similar period of time . However , large amplitude , short time-scale flaring behavior is evident , with trough-to-peak flux changes of about 18 % in approximately 3 days . We present evidence for wavelength-dependent continuum time delays , with the variations at 3000 Å lagging behind those at 1365 Å by about 1 day . These delays may be interpreted as evidence for a stratified continuum reprocessing region , possibly an accretion-disk structure . The Ly \alpha \lambda 1216 emission-line exhibits flux variations of about 1 % amplitude . These variations lag those at 1365Å by { { } _ { < } \atop { } ^ { \sim } } 3 days , and combining this with the line width yields a virial black-hole mass limit of { { } _ { < } \atop { } ^ { \sim } } 8 \times 10 ^ { 6 } M _ { \odot } . We caution , the low amplitude Ly \alpha \lambda 1216 variations may indicate the bulk of the emission region is at larger radii . This scenario affects the veracity of our black hole mass upper limit in an uncertain manner due to the unknown nature of the gas velocity field . Our mass estimate is thus unreliable , however , it is consistent with the independent estimate M \sim 1 \times 10 ^ { 7 } M _ { \odot } of Pounds et al . ( 2001 ) , based on a fluctuation power spectrum analysis of X-ray variability in Akn 564 . The black-hole mass and 5100 Å luminosity of Akn 564 are consistent with the hypothesis that , relative to Seyfert 1 galaxies , NLS1s have lower black hole masses and higher accretion rates . Other strong emission lines , e.g. , C iv \lambda 1549 and He ii \lambda 1640 , are constrained to vary with amplitudes of < 5 % . This low-level of emission-line variability is different from most Seyfert 1 galaxies , which characteristically display variations of \sim 10 % on similar time scales .