The demonstration of magnetic field compression to many tens of megagauss in cylindrical implosions of inertial confinement fusion targets is reported for the first time. The OMEGA laser [T. R. Boehly et al., Opt. Commun. 133, 495 (1997)] was used to implode cylindrical CH targets filled with deuterium gas and seeded with a strong external field (>50 kG) from a specially developed magnetic pulse generator. This seed field was trapped (frozen) in the shock-heated gas fill and compressed by the imploding shell at a high implosion velocity, minimizing the effect of resistive flux diffusion. The magnetic fields in the compressed core were probed via proton deflectrometry using the fusion products from an imploding D 3 He target. Line-averaged magnetic fields between 30 and 40 MG were observed. In the magnetic fusion energy (MFE) concept, a strong magnetic field confines the plasma and reduces the electron thermal conduction to the vessel wall [1]. The magnetic pressure of typical $0:1-MG fields is higher than the total energy density of the plasma (with ¼ 2 0 p=B 2 < 1). MFE plasmas are fully magnetized and characterized by a Hall parameter ! ce > 1 since the modest gyrofrequency ! ce is matched by long collision times . In contrast, typical inertial confinement fusion (ICF) plasmas have collision frequencies higher by 10 to 12 orders of magnitude because of their extreme density. In such systems, thermal conduction losses are a major factor in the energy balance of an implosion. While it can be more difficult, magnetizing the hot spot in ICF implosions can lead to improved gains and to a reduction of the energy required for ignition. A similar approach is used in the magnetized target fusion concept [2], where the fusion burn requires relatively low-implosion velocities, provided there is an adequate magnetic thermal insulation. In ICF implosions, lower implosion velocities lead to higher gains [3]. However, tens of MG are needed to achieve ! ce $ 1 in the hot spot of a typical, direct drive DT ignition target [4] with hot-spot density of $30 g=cc and a temperature of $7 keV. Such a field is higher than both the self-generated magnetic fields (see Ref. [5]) and the external fields that can be generated by coils. Magnetic-flux compression [6] is a viable path to generating tens of MG magnetic fields with adequate size compression of a metal liner driven by high explosives [7,8] or by pulsed power. The latter approach has been pursued by the Z-pinch [9] communities. The results from the first experiments on a new approach that provides very effective flux compression are reported here. The field is compressed by the ablative pressure exerted on an imploding ICF capsule by the driving laser [10]. This approach was proposed in the 1980s [11] as a way to achieve record compressed fields with possible applications for fusion [12] but no laser experiments were performed. There are numerous advantages to this approach as the implosion velocity is high (a few 10 7 cm=s) and the hot plasma is an effective conductor that tra...