Yes it's possible. Sometimes the SSD will come with software (or it will be available for free on the manufacturers website) for doing exactly that. For example the popular Samsung SSD's come with disk cloning software, and also a piece of software called "Samsung Magician" which should be installed after you change the disk over, and automatically optimises Windows for use with the SSD. Other manufacturers provide similar software.
Before you connect the new SSD or perform any cloning, you should set the SATA mode for the AMD SATA controller to AHCI (rather than IDE or RAID) in BIOS and install the Windows AHCI drivers from the Gigabyte website. AHCI gives better performance than IDE mode, particularly with SSDs. Having the drivers pre-installed should ensure that you'll be able to use the SSD in AHCI mode without any issues.
If you are cloning a 250Gb drive to a 250Gb (or larger) drive there should be no issue with drive space. But if you got for example a 120Gb SSD, you might have to make decisions about what you copy over and what you don't.
You can use more than one drive (if there is space in your computer), so you could have the SSD to run the operating system from, and the hard disk to provide extra storage space. You will just have to go into BIOS and change the hard drive priority so that the SSD is booted first.