That Abaqus answer is in kg, meter, sec, so it's not really helpful because leo81 wants kg, millimeters, sec.
BUT, upon further inspection, I was only checking his density number, cause the question wasn't clear. But force should be in mN
Force = kg*mm/s^2 = 1 mN
Stress = kg/mm*s^2 = 1 kPa
Density = kg/mm^3
Therefore a density of 1 g/cm^3 = 1e-6 kg/mm^3, so you'd input 1e-6 into Abaqus.
If you instead used Mg (= 1e3 kg) as the mass, then you get a really nice system of units
Mass = Mg
Length = mm
Time = s
Force = Mg*mm/s^2 = 1 N
Stress = Mg/mm*s^2 = 1 MPa
Density = Mg/mm^3 = 1e12 kg/m^3
So a density of 1 g/cm^3 = 1e-9 Mg/mm^3, so you'd input 1e-9 into Abaqus