In this particular case you don't need to follow the moles, both starting and final point are mass concentration (mass per volume). What is important is that the amount of the substance (here measured as mass) doesn't change. So, if you start with 1 mL you have 1 mg, no matter in what volume it will be diluted, it will be still 1 mg. If you dilute it to 1L you will have 1 mg in 1 L, or 1 mg in 1000 mL, or 1 μg/mL. Do you see where these numbers come from?
No, 1 mg of morphine per 1mL is in no way related to 0.1 M.