We discuss evolution of the magnetic flux density and angular velocity in a molecular cloud core, on the basis of three-dimensional numerical simulations, in which a rotating magnetized cloud fragments and collapses to form a very dense optically thick core of > 5 x 10¹⁰ cm^-3. As the density increases towards the formation of the optically thick core, the magnetic flux density and angular velocity converge towards a single relationship between the two quantities. If the core is magnetically dominated its magnetic flux density approaches 1.5 (n/5 x 10¹⁰ cm^-3)^1/2mG, while if the core is rotationally dominated the angular velocity approaches 2.57 x 10³ (n/5x10¹⁰cm^-3)^1/2 yr^-1, where n is the density of the gas. We also find that the ratio of the angular velocity to the magnetic flux density remains nearly constant until the density exceeds 5 x 10¹⁰ cm^-3. Fragmentation of the very dense core and emergence of outflows from fragments are shown in the subsequent paper.