No. As long as you get your new passport issued before October (26th??) you'll be fine.
www.dhs.gov/interweb/assetlibrary/USVisitRegulation1-5-04.pdf
"
Are Travelers Who Come Under the Visa Waiver Program (VWP) Affected by This Rule?
At this time, travelers who seek to enter under the VWP are not affected by this rule. However, under current law. an alien will not be admitted under the VWP on or after October 26, 2004, without a machine-readable, tamper-resistant passport that meets ICAO biometric standards for photographs, unless his passport is unexpired and was issued prior to that date."
(Emphasis mine..)
The big issue is that the UK is not ready to issue full biometric passports for a while, so passports issued _after_ Oct 26, 2004 but before biometric passports are available will not be valid under the VWP, and a full visa will be needed.
As a Brit living in the US, I'd advise anyone that travels frequently to the US and who holds a UK passport that is due to expire before 2006 to seriously consider renewing it now. Although you'd lose 18 months worth of value from your existing passport, a visa would end up costing you more.
The official date for UK passports to be biometric is currently "mid 2005". Whether that happens or not remains to be seen.