  Using the London approximation within the high field scaling regime, we
calculate the jump in the specific heat $\Delta c$ at the first-order melting
transition of the vortex lattice in YBa$_2$Cu$_3$O$_{7-\delta}$. This has
recently been measured [A. Schilling et al., Phys. Rev. Lett. (78), 4833
(1997)] and reported to be at least 100 times higher than expected from the
fluctuations of field induced vortices alone. We demonstrate how the correct
treatment of the temperature dependence of the model parameters, which are
singular at the mean-field $B_{c2}$ line, leads to good agreement between the
predictions of the London model and the size of the experimental jump. In
addition, we consider the changes in the slopes of the magnetization
$\Delta(\partial M/\partial T)$ and $\Delta(\partial M/\partial H)$ at the
transition. Using continuum anisotropic scaling theory we demonstrate the
consistency of measurements at different angles of the magnetic field with
respect to the crystal c-axis.
