A new anisotropic magnetic fluid is obtained using a colloidal suspension of magnetic particles of nanometric size (ferrofluid) as a component of a swollen lyotropic hexagonal phase. This doped hexagonal system exhibits specific behaviors when submitted to a magnetic field of weak intensity. The field-induced instabilities are described and interpreted; they result from a high anisotropy of the magnetic susceptibility of the medium, which is measured. It is finally shown that the magnetic properties of the doped hexagonal phase allows one a determination of the compression modulus of the system.