calculus and analysis - Why don't products of Dirac deltas integrate correctly?

Bug introduced in 10.0.0 and fixed in 10.0.1

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The integral $\int \int \ \delta(x) \delta(y) \ dx dy=1$ evaluates to 0 in Mathematica

Integrate[
    DiracDelta[x]*DiracDelta[y],
    {x, -Infinity, Infinity}, 
    {y, -Infinity, Infinity}
]

returns 0 even though

Integrate[
    DiracDelta[x],
    {x, -Infinity, Infinity}
]

correctly returns 1. What's going on?

Edit: This concerns Version 10.0.0.0 installed on Ubuntu 14.04. The notebook contains no other commands. Clear[x,y] does not affect the outcome.