Now that is interesting. I tried several DVDs, and all worked fine. I tried Ice Age again, and it worked fine as well. Very weird. So I guess my problem is "fixed." Just in case anything has changed, below is my latest system information.
Media Center Registered 9.1.274 -- C:\Program Files\J River\Media Center\
Microsoft Windows XP Workstation 5.1 Service Pack 1 (Build 2600)
AMD Athlon 1395 MHz MMX / Memory: Total - 523 MB, Free - 246 MB
Internet Explorer: 6.0.2800.1106 / ComCtl32.dll: 5.82 (xpsp1.020828-1920) / Shlwapi.dll: 6.00.2800.1226 / Shell32.dll: 6.00.2800.1233 (xpsp2.030604-1804) / wnaspi32.dll: 4.01 (978) , ASPI for Win32 (95/NT) DLL, Copyright © 1989-1996 Adaptec, Inc. / Aspi32.sys: N/A
Ripping / Drive D: Copy mode:ModeBurstBigBuffer CD Type:Auto Read speed:Max
Drive E: Copy mode:ModeBurstBigBuffer CD Type:Auto Read speed:Max
Drive F: Copy mode:ModeBurstBigBuffer CD Type:Auto Read speed:Max
Drive I: Copy mode:ModeBurstBigBuffer CD Type:Auto Read speed:Max
Digital playback: Yes / Use YADB: Yes / Get cover art: No / Calc replay gain: Yes / Copy volume: 32767
Eject after ripping: Yes / Play sound after ripping: No
Burning / Drive D: AOPEN CD-RW CRW1232 A Addr: 1:0:0 Speed:2 MaxSpeed:12 BurnProof:No
Drive E: SONY DVD RW DRU-500A Addr: 1:1:0 Speed:4 MaxSpeed:4 BurnProof:Yes
Test mode: No / Eject after writing: Yes / Direct decoding: No / Write CD-Text: No
Use playback settings: No / Normalization: None