Impose the color for one discrete value above a color predefined gradient

Adding the option

ColorRules -> {0 -> White} 

to ArrayPlot works.

Sorry I find the answer 1 min after I ask the question...

Not all plot functions accept ColorRules, so it is good to know how to construct a custom color function anyway. For this example:

cf = If[# == 0, White, "Rainbow" ~Blend~ #] &;

ArrayPlot[a, ColorFunction -> cf]

More specified colors can be applied with Piecewise. Custom color functions can also apply to multiple dimensions:

cf = Piecewise[{
     {Green, 0.2 < #2 < 0.3},
     {Cyan, 0.7 < #2 < 0.8},
     {Yellow, 0.2 < #1 < 0.3},
     {Black, 0.7 < #1 < 0.8}
     }, Blend["Rainbow", #]] &;

Plot[TriangleWave[x], {x, 0, 5},
 PlotStyle -> AbsoluteThickness[5], ColorFunction -> cf, PlotPoints -> 5000]

You should also see MeshFunctions, MeshStyle, etc. for such things, which would not require such an extreme PlotPoints value, but I wished to make a point.

One could also use Lighter[] for the purpose:

ArrayPlot[a, ColorFunction -> (Lighter[ColorData["Rainbow", #], Boole[# == 0]] &)]