New jump conditions for an immersed interface method (IIM) for the solution of the Navier--Stokes equations with continuous viscosity are presented. The difference between the new jump conditions and those in some of the existing IIM literature (for example, D. V. Le, B. C. Khoo and J. Peraire [J. Comput. Phys., 220 (2006), pp. 109--138], Z. Li and M.-C. Lai [J. Comput. Phys., 171 (2001), pp. 822--842]) is found in those pertaining to a scalar function $\varphi$ from which the fluid pressure is calculated and which satisfies a Poisson equation in the PmII projection scheme of Brown, Cortez, and Minion [J. Comput. Phys. 168 (2001), pp. 464--499]. In particular, it is shown that the jump terms for an intermediate velocity $\mathbf{\mathit{v}}^*$ in the first step of the projection scheme arise naturally as being those for the fluid velocity $\mathbf{\mathit{v}}$ if it is assumed that $\varphi$ is twice continuously differentiable throughout the flow domain $\Omega$. Conversely, it is proved that if all the ...