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If you're waiting on a vital email, or just a compulsive Web surfer, then Whitby is a mixed bag in 2007.
The Tourist Information will point you in the direction of the library (2 terminals) or the cafe (Java Cafe?) opposite the supermarket. Neither is satisfactory - the library terminals can be busy and the library closes early (so zero evening access), and the Java Cafe (2 terminals) is a total ripoff at £5 per hour (although you can pay in smaller instalments of time).
However, the wi-fi cafe in the old cinema (next to the bus station) is the place to go. Friendly staff, clean and tidy, a good selection of teas/coffee cakes and confections, access to toilets, around 12 terminals, open until 8 or 9pm, great broadband prices (£1 for 40mins) - there is no contest. Enjoy.