Hi Ted! The problem is that DiracDelta is not really a function, but a positive measure (or, more generally a Schwartz's distribution). So, we must use a sequence of functions converging towards DiracDelta to obtain the Laplace Transform.
Try for instance this sequence:
f[t_, r_] := If[t > r, 0, 1/r]
Plot[Table[f[t, r], {r, 1/10, 2, 1/10}], {t, 0, 3}, PlotRange -> All]
Then, compute the Laplace transform, and its limit when the width converge to zero:
Integrate[f[t, r] E^(-s*t), {t, 0, \[Infinity]}, Assumptions -> r > 0]
Limit[%, r -> 0]
You will obtain 1.