If you were caught doing this, and the government wanted to nail you for this, they certainly could.
See:
Hazard Class 6 consists of two divisions: Division 6.1 includes toxic substances, poisons, and irritating material. Examples of Division 6.1 materials (not all of which are mailable) include bromobenzyl cyanide, methyl bromide, motor fuel anti-knock mixtures, and tear gas. Division 6.2 includes infectious substances. Examples of Division 6.2 materials include infectious substances, biological products, regulated medical waste, sharps medical waste, used health care products, and forensic materials.
But specifically:
Infectious substance means a material known or reasonably expected to contain a pathogen. A pathogen is a microorganism that can cause disease in humans or animals. Examples of pathogens include bacteria, viruses, fungi, and other infectious agents. An infectious substance must be assigned to one of the following two packaging categories:
...BUT! There's hope for you yet!
If you do it right, you could protect yourself by following the alternative governing procedures regarding the transmission of naughty substances:
346.326 Exempt Human or Animal Specimens
Exempt human or animal specimens as defined in 346.12d are not subject to regulation as hazardous materials but when presented for mailing must be triple-packaged in leakproof (for liquids) or siftproof (for solids) primary receptacles. Sufficient cushioning and absorbent materials must surround each primary receptacle containing liquid. Secondary containers for liquids must be leakproof. Secondary containers for solids must be siftproof. The primary and secondary packaging must be enclosed in a rigid outer shipping container. A single primary receptacle must not contain more than 500 ml of a liquid specimen or 500 grams of a solid specimen. Two or more primary receptacles whose combined volume does not exceed 500 ml (for liquids) or 500 grams (for solids) may be enclosed in a single secondary container. The secondary container cannot serve as the outer shipping container. The secondary container must be marked with the international biohazard symbol shown in Exhibit 346.321. The secondary container must be securely and snugly enclosed in a fiberboard box or container of equivalent strength that serves as the outer shipping container. A shipping paper is not required. The outer shipping container must be marked on the address side with the words “Exempt human specimen” or “Exempt animal specimen,” as appropriate. In addition, at least one surface of the outer packaging must have a minimum dimension of 3.9 inches by 3.9 inches (100 mm by 100 mm). Exempt human and animal specimens are mailable as Priority Mail Express, Priority Mail, First-Class Mail, First-Class Package Service, Parcel Select, or Standard Post.