Possible, yes. We were just having a discussion somewhere on migration tools to migrate a properly licensed OS to a new drive or a new machine. Is the OEM license something you installed or did it come with the machine?
There are various migration tools for this. You shouldn't do a straight clone because boundary alignments need to be changed between an HDD and an SSD. Also, if the disk controllers on the motherboard were in IDE mode, they should be put into AHCI mode and the migration tool will tweak the setting on the way over.
You need a migration tool and a way to attach both drives at once. Is this a laptop, or can you install both drives at the same time in a tower case? I would personally choose Acronis True Image to do this, but I'm quite behind the time. The method that I know involved backing up the original disk to an image and restoring the image to the SSD, but I know that some tools can do it in one step. Hopefully, someone with more recent experience will step in here and name a better tool.