  Ground-state and finite-temperature properties of $S=1/2$ Heisenberg ladders
with a ferromagnetic leg, an antiferromagnetic leg, and antiferromagnetic rungs
are studied. It is shown that a partial ferrimagnetic phase extends over a wide
parameter range in the ground state. The numerical results are supported by an
analytical calculation based on a mapping onto the nonlinear $\sigma$ model and
a perturbation calculation from the strong-rung limit. It is shown that the
partial ferrimagnetic state is a spontaneously magnetized Tomonaga--Luttinger
liquid with incommensurate magnetic correlation, which is confirmed by a DMRG
calculation. The finite-temperature magnetic susceptibility is calculated using
the thermal pure quantum state method. It is suggested that the susceptibility
diverges as $T^{-2}$ in the ferrimagnetic phases as in the case of
ferromagnetic Heisenberg chains.
