The isoscalar monopole response has been measured in the unstable nucleus 68 Ni using inelastic alpha scattering at 50A MeV in inverse kinematics with the active target MAYA at GANIL. The isoscalar giant monopole resonance (ISGMR) centroid was determined to be 21.1 AE 1.9 MeV and indications for a soft monopole mode are provided for the first time at 12.9 AE 1.0 MeV. Analysis of the corresponding angular distributions using distorted-wave-born approximation with random-phase approximation transition densities indicates that the L ¼ 0 multipolarity dominates the cross section for the ISGMR and significantly contributes to the low-energy mode. The L ¼ 0 part of this low-energy mode, the soft monopole mode, is dominated by neutron excitations. This demonstrates the relevance of inelastic alpha scattering in inverse kinematics in order to probe both the ISGMR and isoscalar soft modes in neutron-rich nuclei. Measurements in exotic nuclei have yielded an abundance of new information in recent years. For instance, a more general view on magicity has emerged in the past decade [1]. It has also been suggested that measurements of giant resonances in exotic nuclei could help clarify another important problem related to nuclear incompressibility [2]: despite significant progress in our understanding of the nuclear incompressibility, one cannot converge towards a value that is more accurate than the 10%-20% level. In other words, it seems difficult to single out an energy density functional that can give a comprehensive description of the ISGMR, especially in keeping with new data on isotopic chains including open-shell systems and on neutron-rich nuclei [3][4][5][6]. This can be seen in the recent theoretical works on the subject [7][8][9][10][11][12]. The measurement of the ISGMR in exotic neutron-rich nuclei is, therefore, of paramount importance. Such measurements in exotic nuclei would significantly improve our understanding of nuclear incompressibility as was the case for nuclear magicity in the past decade. In addition, a soft monopole mode has been predicted around 14 MeV in neutron-rich nuclei by several relativistic and nonrelativistic models [13][14][15], but it has never been observed, due to the difficulty in measuring the monopole response in nuclei far from stability. The soft monopole mode is predicted to be noncollective and its observation could bring valuable information on spin-orbit splitting [16]. It is therefore essential to measure the isoscalar (in-phase motion of protons and neutrons) monopole response in neutron-rich nuclei.The measurement of giant resonances in unstable nuclei is a particularly challenging task and has, until now, been mainly limited to the isovector (protons and neutrons out of phase) giant dipole resonance in neutron-rich radioactive O, Ne, Sn isotopes, and in 68 Ni [17]. Such measurements provided the first evidence for soft isovector dipole modes in unstable nuclei. These modes are often reanalyzed, showing a nontrivial pattern, departing from the simple pygmy resonance pictu...