If it is a dilute solution in water, you could assume it is nearly the same density as water, 1 kg/L. If this is a reasonable assumption, then it is the same numeric value in mg/kg and mg/L. Otherwise, you must multiply by the actual mixture density in kg/L.

Assuming this, note that 1000 µg = 1 mg, and 1000 mL = 1 L, so the 1000's cancel out, and 1 mg/L = 1 µg/mL