We report on 49 fast-mode forward shocks propagating inside coronal mass ejections (CMEs) as measured by Wind and ACE at 1 AU from 1997 to 2006. Compared to typical CME-driven shocks, these shocks propagate in different upstream conditions, where the median upstream Alfv{é}n speed is 85 km s$^{-1}$, the proton $β = 0.08$ and the magnetic field strength is 8 nT. These shocks are fast with a median speed of 590 km s$^{-1}$ but weak with a median Alfv{é}nic Mach number of 1.9. They typically compress the magnetic field and density by a factor of 2-3. The most extreme upstream conditions found were a fast magnetosonic speed of 230 km s$^{-1}$, a plasma $β$ of 0.02, upstream solar wind speed of 740 km s$^{-1}$ and density of 0.5 cm$^{-3}$. Nineteen of these complex events were associated with an intense geomagnetic storm (peak Dst under $-100$ nT) within 12 hours of the shock detection at Wind, and fifteen were associated with a drop of the storm-time Dst index of more than 50 nT between 3 and 9 hours after shock detection. We also compare them to a sample of 45 shocks propagating in more typical upstream conditions. We show the average property of these shocks through a superposed epoch analysis, and we present some analytical considerations regarding the compression ratios of shocks in low $β$ regimes. As most of these shocks are measured in the back half of a CME, we conclude that about half the shocks may not remain fast-mode shocks as they propagate through an entire CME due to the large upstream and magnetosonic speeds. ; Comment: Accepted to JGR, 17 pages, 2 tables, 10 figures