Supercomputer companies such as Cray, Silicon Graphics, and SRC Computers now offer reconfigurable computer (RC) systems that combine general-purpose processors (GPPs) with field-programmable gate arrays (FPGAs). The FPGAs can be programmed to become, in effect, application-specific processors. These exciting supercomputers allow end-users to create custom computing architectures aimed at the computationally intensive parts of each problem. This report describes a parameterized, parallelized, deeply pipelined, dual-FPGA, IEEE-754 64-bit floating-point design for accelerating the conjugate gradient (CG) iterative method on an FPGA-augmented RC. The FPGA-based elements are developed via a hybrid approach that uses a high-level language (HLL)-to-hardware description language (HDL) compiler in conjunction with custombuilt, VHDL-based, floating-point components. A reference version of the design is implemented on a contemporary RC. Actual run time performance data compare the FPGAaugmented CG to the software-only version and show that the FPGA-based version runs 1.3 times faster than the software version. Estimates show that the design can achieve a 4 fold speedup on a next-generation RC.