Mithril and Adamantine are not magical. They are simply another type of metal that has very powerful properties compared to ordinary steel.
However, I believe Adamantine weapons do bypass DR. It provides an enhancement bonus of +1 or +2 as seen on the table on the bottom right hand side of page 242 of the DMG. This bonus is labeled an enhancement bonus.
A look on table 3-13 on page 74 of the damage tells us that an enhancement bonus is what is needed to get past DR.
Additionally, the second bulleted item under the Damage Reduction section beginning on page 73 states that "a certain type of weapon--usually a magic weapon--can overcome this reduction." This indicates to me that a weapon does not necessarily have to be magical in order to bypass DR.